| L(s) = 1 | + 8·7-s − 18·9-s + 56·17-s − 112·23-s − 12·25-s − 552·31-s − 1.17e3·41-s − 1.42e3·47-s − 852·49-s − 144·63-s + 432·71-s − 728·73-s − 2.15e3·79-s + 243·81-s − 4.61e3·89-s − 3.80e3·97-s − 1.40e3·113-s + 448·119-s + 2.60e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s − 1.00e3·153-s + 157-s + ⋯ |
| L(s) = 1 | + 0.431·7-s − 2/3·9-s + 0.798·17-s − 1.01·23-s − 0.0959·25-s − 3.19·31-s − 4.47·41-s − 4.41·47-s − 2.48·49-s − 0.287·63-s + 0.722·71-s − 1.16·73-s − 3.06·79-s + 1/3·81-s − 5.49·89-s − 3.97·97-s − 1.16·113-s + 0.345·119-s + 1.95·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s − 0.532·153-s + 0.000508·157-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1108602129\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1108602129\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| good | 5 | $D_4\times C_2$ | \( 1 + 12 T^{2} + 15926 T^{4} + 12 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $D_{4}$ | \( ( 1 - 4 T + 450 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 2604 T^{2} + 3702326 T^{4} - 2604 p^{6} T^{6} + p^{12} T^{8} \) |
| 13 | $C_2^2$ | \( ( 1 - 4390 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 28 T + 1382 T^{2} - 28 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 19244 T^{2} + 174632406 T^{4} - 19244 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 56 T + 16478 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 6068 T^{2} + 357799638 T^{4} - 6068 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 276 T + 66866 T^{2} + 276 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 32492 T^{2} + 5072441334 T^{4} - 32492 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 588 T + 223318 T^{2} + 588 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 81100 T^{2} + 13216498038 T^{4} - 81100 p^{6} T^{6} + p^{12} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 712 T + 333422 T^{2} + 712 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 + 14380 T^{2} + 44087902518 T^{4} + 14380 p^{6} T^{6} + p^{12} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 401388 T^{2} + 84923349878 T^{4} - 401388 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_4\times C_2$ | \( 1 - 481292 T^{2} + 127804965078 T^{4} - 481292 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_4\times C_2$ | \( 1 + 346036 T^{2} + 144908686422 T^{4} + 346036 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 216 T + 718846 T^{2} - 216 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 73 | $D_{4}$ | \( ( 1 + 364 T + 162198 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 1076 T + 1221522 T^{2} + 1076 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 232076 T^{2} + 279958929942 T^{4} - 232076 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 2308 T + 2645654 T^{2} + 2308 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 1900 T + 2631846 T^{2} + 1900 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−6.88853917718265527250879439818, −6.86336475523759520123817084969, −6.85014959450185122059775805108, −6.14320051641911930382212744595, −6.07090977745336851934674071614, −5.75099620502701908258058214736, −5.69095043651640748028977447996, −5.21189520776515269528237392362, −5.14560831625633223604511527619, −5.01424985489514144844226054363, −4.68422603463647820335168843557, −4.31686460986646198699751250760, −4.03254613008294940129151712470, −3.84717752322000195732379627809, −3.30313547788746285428742807089, −3.22158435606585257313909049117, −3.11404583536836708047464391349, −2.95465838085148444474409774922, −2.13977799775418896937609461038, −1.95175944372725013555286923428, −1.58635461835675015776279794880, −1.49162409661978782059331074832, −1.34931043419613469191738189401, −0.17946107015099644389741826970, −0.12934971004946441967617708053,
0.12934971004946441967617708053, 0.17946107015099644389741826970, 1.34931043419613469191738189401, 1.49162409661978782059331074832, 1.58635461835675015776279794880, 1.95175944372725013555286923428, 2.13977799775418896937609461038, 2.95465838085148444474409774922, 3.11404583536836708047464391349, 3.22158435606585257313909049117, 3.30313547788746285428742807089, 3.84717752322000195732379627809, 4.03254613008294940129151712470, 4.31686460986646198699751250760, 4.68422603463647820335168843557, 5.01424985489514144844226054363, 5.14560831625633223604511527619, 5.21189520776515269528237392362, 5.69095043651640748028977447996, 5.75099620502701908258058214736, 6.07090977745336851934674071614, 6.14320051641911930382212744595, 6.85014959450185122059775805108, 6.86336475523759520123817084969, 6.88853917718265527250879439818
