| L(s) = 1 | + 4·3-s + 8·5-s + 12·7-s − 6·9-s + 4·11-s + 8·13-s + 32·15-s + 48·21-s + 32·25-s − 40·27-s + 40·29-s − 40·31-s + 16·33-s + 96·35-s + 40·37-s + 32·39-s + 32·41-s − 48·45-s + 4·47-s + 72·49-s − 80·53-s + 32·55-s − 56·59-s − 296·61-s − 72·63-s + 64·65-s + 84·67-s + ⋯ |
| L(s) = 1 | + 4/3·3-s + 8/5·5-s + 12/7·7-s − 2/3·9-s + 4/11·11-s + 8/13·13-s + 2.13·15-s + 16/7·21-s + 1.27·25-s − 1.48·27-s + 1.37·29-s − 1.29·31-s + 0.484·33-s + 2.74·35-s + 1.08·37-s + 0.820·39-s + 0.780·41-s − 1.06·45-s + 4/47·47-s + 1.46·49-s − 1.50·53-s + 0.581·55-s − 0.949·59-s − 4.85·61-s − 8/7·63-s + 0.984·65-s + 1.25·67-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
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(8.388084642\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.388084642\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 8 T + 8 p T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} \) |
| good | 3 | $D_{4}$ | \( ( 1 - 2 T + p^{2} T^{2} - 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 5 | $D_4\times C_2$ | \( 1 - 8 T + 32 T^{2} - 224 T^{3} + 1559 T^{4} - 224 p^{2} T^{5} + 32 p^{4} T^{6} - 8 p^{6} T^{7} + p^{8} T^{8} \) |
| 7 | $D_4\times C_2$ | \( 1 - 12 T + 72 T^{2} - 744 T^{3} + 7519 T^{4} - 744 p^{2} T^{5} + 72 p^{4} T^{6} - 12 p^{6} T^{7} + p^{8} T^{8} \) |
| 11 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} - 172 T^{3} - 2386 T^{4} - 172 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 - 418 T^{2} + 197763 T^{4} - 418 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $C_2^3$ | \( 1 + 232162 T^{4} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 56 p T^{2} + 857778 T^{4} - 56 p^{5} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 20 T + 1532 T^{2} - 20 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 + 40 T + 800 T^{2} + 39240 T^{3} + 1924322 T^{4} + 39240 p^{2} T^{5} + 800 p^{4} T^{6} + 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 37 | $D_4\times C_2$ | \( 1 - 40 T + 800 T^{2} - 46560 T^{3} + 2667767 T^{4} - 46560 p^{2} T^{5} + 800 p^{4} T^{6} - 40 p^{6} T^{7} + p^{8} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 32 T + 512 T^{2} - 57248 T^{3} + 6389378 T^{4} - 57248 p^{2} T^{5} + 512 p^{4} T^{6} - 32 p^{6} T^{7} + p^{8} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 - 6334 T^{2} + 16708731 T^{4} - 6334 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4 T + 8 T^{2} + 1736 T^{3} - 6608737 T^{4} + 1736 p^{2} T^{5} + 8 p^{4} T^{6} - 4 p^{6} T^{7} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 40 T + 5768 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $D_4\times C_2$ | \( 1 + 56 T + 1568 T^{2} - 2632 T^{3} - 12442366 T^{4} - 2632 p^{2} T^{5} + 1568 p^{4} T^{6} + 56 p^{6} T^{7} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 148 T + 12878 T^{2} + 148 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 84 T + 3528 T^{2} + 155316 T^{3} - 33332642 T^{4} + 155316 p^{2} T^{5} + 3528 p^{4} T^{6} - 84 p^{6} T^{7} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 4 p T + 8 p^{2} T^{2} + 57112 p T^{3} + 322400399 T^{4} + 57112 p^{3} T^{5} + 8 p^{6} T^{6} + 4 p^{7} T^{7} + p^{8} T^{8} \) |
| 73 | $C_2^3$ | \( 1 + 18164482 T^{4} + p^{8} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 32 T + 11528 T^{2} + 32 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 52 T + 1352 T^{2} - 271804 T^{3} + 51880814 T^{4} - 271804 p^{2} T^{5} + 1352 p^{4} T^{6} - 52 p^{6} T^{7} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 200 T + 20000 T^{2} - 1908200 T^{3} + 179436962 T^{4} - 1908200 p^{2} T^{5} + 20000 p^{4} T^{6} - 200 p^{6} T^{7} + p^{8} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 + 68 T + 2312 T^{2} - 801924 T^{3} - 171375106 T^{4} - 801924 p^{2} T^{5} + 2312 p^{4} T^{6} + 68 p^{6} T^{7} + p^{8} T^{8} \) |
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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
−8.939640644382175949935818941768, −8.483033003701196719279952232486, −8.359367747273525374378072002871, −8.038834001341250591921380609424, −8.029199470191051512799580969312, −7.47299571287991868020228713956, −7.35937318323255324956295332599, −7.05667785329486412192249512343, −6.55329699642273357664447936467, −6.06242551557550310959294263968, −6.03966652139760191061086577355, −5.82463654990154189401131821599, −5.68097230524140416696318436766, −5.05880189258667846497430071209, −4.60975798894212141859831732167, −4.54873073394124977178060928299, −4.43304887344879420687733212849, −3.64975476727245367558399816993, −3.03530712675039977622326697430, −2.97863901726508006466120852053, −2.89480574666267204274000225367, −1.97067577102024815637266614624, −1.81282408117303132539035799773, −1.59837840763388464110903497738, −0.73857316637235001494866662507,
0.73857316637235001494866662507, 1.59837840763388464110903497738, 1.81282408117303132539035799773, 1.97067577102024815637266614624, 2.89480574666267204274000225367, 2.97863901726508006466120852053, 3.03530712675039977622326697430, 3.64975476727245367558399816993, 4.43304887344879420687733212849, 4.54873073394124977178060928299, 4.60975798894212141859831732167, 5.05880189258667846497430071209, 5.68097230524140416696318436766, 5.82463654990154189401131821599, 6.03966652139760191061086577355, 6.06242551557550310959294263968, 6.55329699642273357664447936467, 7.05667785329486412192249512343, 7.35937318323255324956295332599, 7.47299571287991868020228713956, 8.029199470191051512799580969312, 8.038834001341250591921380609424, 8.359367747273525374378072002871, 8.483033003701196719279952232486, 8.939640644382175949935818941768
