| L(s) = 1 | + 16·5-s − 16·13-s + 8·17-s + 108·25-s + 80·29-s − 16·37-s + 72·41-s + 180·49-s + 144·53-s + 176·61-s − 256·65-s + 40·73-s + 128·85-s − 136·89-s − 264·97-s + 208·101-s + 176·109-s − 328·113-s + 132·121-s + 432·125-s + 127-s + 131-s + 137-s + 139-s + 1.28e3·145-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | + 16/5·5-s − 1.23·13-s + 8/17·17-s + 4.31·25-s + 2.75·29-s − 0.432·37-s + 1.75·41-s + 3.67·49-s + 2.71·53-s + 2.88·61-s − 3.93·65-s + 0.547·73-s + 1.50·85-s − 1.52·89-s − 2.72·97-s + 2.05·101-s + 1.61·109-s − 2.90·113-s + 1.09·121-s + 3.45·125-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 8.82·145-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(26.36255998\) |
| \(L(\frac12)\) |
\(\approx\) |
\(26.36255998\) |
| \(L(2)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $D_{4}$ | \( ( 1 - 8 T + 42 T^{2} - 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 - 90 T^{2} + p^{4} T^{4} )^{2} \) |
| 11 | $D_4\times C_2$ | \( 1 - 12 p T^{2} + 9062 T^{4} - 12 p^{5} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 8 T + 258 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_{4}$ | \( ( 1 - 4 T + 198 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 1092 T^{2} + 534182 T^{4} - 1092 p^{4} T^{6} + p^{8} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 516 T^{2} + 998 p^{2} T^{4} - 516 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 - 40 T + 1482 T^{2} - 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 31 | $D_4\times C_2$ | \( 1 - 1012 T^{2} + 112422 T^{4} - 1012 p^{4} T^{6} + p^{8} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 8 T + 1218 T^{2} + 8 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 36 T + 3302 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 900 T^{2} + 6425702 T^{4} - 900 p^{4} T^{6} + p^{8} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 2116 T^{2} + 7339782 T^{4} - 2116 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 72 T + 5738 T^{2} - 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 5762 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 - 88 T + 8994 T^{2} - 88 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 9156 T^{2} + 41992742 T^{4} - 9156 p^{4} T^{6} + p^{8} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 22417862 T^{4} - 132 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{4} \) |
| 79 | $D_4\times C_2$ | \( 1 - 6516 T^{2} + 24592550 T^{4} - 6516 p^{4} T^{6} + p^{8} T^{8} \) |
| 83 | $D_4\times C_2$ | \( 1 - 13636 T^{2} + 114638502 T^{4} - 13636 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 68 T + 15462 T^{2} + 68 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 132 T + 21638 T^{2} + 132 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−6.13714210793912672941375011774, −5.75079821356121455830113164756, −5.66107086583649003649566923687, −5.61587168974909389883533178536, −5.55336875433421783274656298646, −5.43105341259992893026167513796, −5.05188527473606015987708677976, −4.81435311677956656673880705516, −4.56146552710111278721925219465, −4.23957768000489075073527152297, −4.06059976865255309693576894076, −4.05351957875723587519844221798, −3.70615420339625876684637421755, −2.99931309991569724820960171640, −2.89747054395054185065174061937, −2.85764398403672776530846142503, −2.71622727099446077455180655127, −2.13304077810048205479619027763, −2.05514014509180295184303632940, −1.98629627249153456701844631296, −1.85437999701848035458790860301, −1.03484675878350322318138120779, −1.00723884004907155791989704403, −0.63287447952298377357710948555, −0.60758852781387371657047967876,
0.60758852781387371657047967876, 0.63287447952298377357710948555, 1.00723884004907155791989704403, 1.03484675878350322318138120779, 1.85437999701848035458790860301, 1.98629627249153456701844631296, 2.05514014509180295184303632940, 2.13304077810048205479619027763, 2.71622727099446077455180655127, 2.85764398403672776530846142503, 2.89747054395054185065174061937, 2.99931309991569724820960171640, 3.70615420339625876684637421755, 4.05351957875723587519844221798, 4.06059976865255309693576894076, 4.23957768000489075073527152297, 4.56146552710111278721925219465, 4.81435311677956656673880705516, 5.05188527473606015987708677976, 5.43105341259992893026167513796, 5.55336875433421783274656298646, 5.61587168974909389883533178536, 5.66107086583649003649566923687, 5.75079821356121455830113164756, 6.13714210793912672941375011774
