| L(s) = 1 | − 4·3-s + 6·5-s − 16·7-s + 10·9-s + 8·11-s − 24·15-s − 44·19-s + 64·21-s − 12·23-s + 18·25-s + 16·27-s + 48·29-s + 24·31-s − 32·33-s − 96·35-s − 26·37-s − 116·41-s + 168·43-s + 60·45-s − 80·47-s + 128·49-s + 212·53-s + 48·55-s + 176·57-s + 168·59-s + 22·61-s − 160·63-s + ⋯ |
| L(s) = 1 | − 4/3·3-s + 6/5·5-s − 2.28·7-s + 10/9·9-s + 8/11·11-s − 8/5·15-s − 2.31·19-s + 3.04·21-s − 0.521·23-s + 0.719·25-s + 0.592·27-s + 1.65·29-s + 0.774·31-s − 0.969·33-s − 2.74·35-s − 0.702·37-s − 2.82·41-s + 3.90·43-s + 4/3·45-s − 1.70·47-s + 2.61·49-s + 4·53-s + 0.872·55-s + 3.08·57-s + 2.84·59-s + 0.360·61-s − 2.53·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9843046716\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9843046716\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - p^{2} T^{2} + p^{4} T^{4} \) |
| good | 3 | $D_4\times C_2$ | \( 1 + 4 T + 2 p T^{2} - 32 T^{3} - 125 T^{4} - 32 p^{2} T^{5} + 2 p^{5} T^{6} + 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 5 | $D_4\times C_2$ | \( 1 - 6 T + 18 T^{2} - 96 T^{3} + 431 T^{4} - 96 p^{2} T^{5} + 18 p^{4} T^{6} - 6 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $C_2^3$ | \( 1 + 16 T + 128 T^{2} + 480 T^{3} + 1439 T^{4} + 480 p^{2} T^{5} + 128 p^{4} T^{6} + 16 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 - 8 T + 272 T^{2} - 72 p T^{3} + 263 p^{2} T^{4} - 72 p^{3} T^{5} + 272 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 577 T^{2} + 249408 T^{4} + 577 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_4\times C_2$ | \( 1 + 44 T + 1160 T^{2} + 22440 T^{3} + 421151 T^{4} + 22440 p^{2} T^{5} + 1160 p^{4} T^{6} + 44 p^{6} T^{7} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 + 12 T + 1018 T^{2} + 11640 T^{3} + 686451 T^{4} + 11640 p^{2} T^{5} + 1018 p^{4} T^{6} + 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 48 T + 121 T^{2} - 24048 T^{3} + 2120544 T^{4} - 24048 p^{2} T^{5} + 121 p^{4} T^{6} - 48 p^{6} T^{7} + p^{8} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 24 T + 288 T^{2} - 22488 T^{3} + 1755362 T^{4} - 22488 p^{2} T^{5} + 288 p^{4} T^{6} - 24 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 + 26 T + 845 T^{2} + 44382 T^{3} + 5288 T^{4} + 44382 p^{2} T^{5} + 845 p^{4} T^{6} + 26 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 + 116 T + 3413 T^{2} - 195024 T^{3} - 16796548 T^{4} - 195024 p^{2} T^{5} + 3413 p^{4} T^{6} + 116 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 168 T + 358 p T^{2} - 1005648 T^{3} + 49808787 T^{4} - 1005648 p^{2} T^{5} + 358 p^{5} T^{6} - 168 p^{6} T^{7} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 + 80 T + 3200 T^{2} + 223440 T^{3} + 15260642 T^{4} + 223440 p^{2} T^{5} + 3200 p^{4} T^{6} + 80 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 2 p T + 7659 T^{2} - 2 p^{3} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 - 168 T + 7200 T^{2} + 578496 T^{3} - 72286897 T^{4} + 578496 p^{2} T^{5} + 7200 p^{4} T^{6} - 168 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 22 T - 6311 T^{2} + 14234 T^{3} + 30525220 T^{4} + 14234 p^{2} T^{5} - 6311 p^{4} T^{6} - 22 p^{6} T^{7} + p^{8} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 - 84 T + 14760 T^{2} - 980208 T^{3} + 95976863 T^{4} - 980208 p^{2} T^{5} + 14760 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 60 T + 3816 T^{2} + 337872 T^{3} + 4654991 T^{4} + 337872 p^{2} T^{5} + 3816 p^{4} T^{6} + 60 p^{6} T^{7} + p^{8} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 126 T + 7938 T^{2} - 171360 T^{3} - 12053761 T^{4} - 171360 p^{2} T^{5} + 7938 p^{4} T^{6} - 126 p^{6} T^{7} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 8 T + 7698 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 + 56 T + 1568 T^{2} + 395640 T^{3} + 99797474 T^{4} + 395640 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 190 T + 9554 T^{2} - 1588848 T^{3} - 256754785 T^{4} - 1588848 p^{2} T^{5} + 9554 p^{4} T^{6} + 190 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 222 T + 35730 T^{2} - 4074960 T^{3} + 437335871 T^{4} - 4074960 p^{2} T^{5} + 35730 p^{4} T^{6} - 222 p^{6} T^{7} + p^{8} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.939355972297154046067123020695, −8.568806913662558134796898378637, −8.558686481792444059524701989761, −8.036184910942505993053001942720, −7.78392045687872189226454116559, −7.17864406911898652155726850746, −6.70286759187650182706960539346, −6.66249930911884730093264080384, −6.62756759315893062687078284965, −6.60202515289970832062574799126, −6.22042625421287930881556005279, −5.67957325268525757139977271640, −5.57345020182295601774791586258, −5.27908305036806156543306970417, −5.02237830133679010092542620000, −4.45397512041352412891843045032, −3.97568154879292949635036450549, −3.87464882768915588269720562807, −3.77427001303207315099864474729, −2.66544138662814082682999831138, −2.60760711728503046147782910939, −2.51245735836311616638017363550, −1.55832155187435911997665296612, −0.980790230966550365201427073331, −0.37698746677761546622943143503,
0.37698746677761546622943143503, 0.980790230966550365201427073331, 1.55832155187435911997665296612, 2.51245735836311616638017363550, 2.60760711728503046147782910939, 2.66544138662814082682999831138, 3.77427001303207315099864474729, 3.87464882768915588269720562807, 3.97568154879292949635036450549, 4.45397512041352412891843045032, 5.02237830133679010092542620000, 5.27908305036806156543306970417, 5.57345020182295601774791586258, 5.67957325268525757139977271640, 6.22042625421287930881556005279, 6.60202515289970832062574799126, 6.62756759315893062687078284965, 6.66249930911884730093264080384, 6.70286759187650182706960539346, 7.17864406911898652155726850746, 7.78392045687872189226454116559, 8.036184910942505993053001942720, 8.558686481792444059524701989761, 8.568806913662558134796898378637, 8.939355972297154046067123020695
