| L(s) = 1 | − 2·5-s − 5·13-s + 5·17-s − 19-s − 4·23-s − 25-s − 10·29-s − 10·31-s + 2·37-s + 10·43-s + 8·47-s − 7·49-s − 53-s − 5·59-s − 10·61-s + 10·65-s − 8·67-s − 3·71-s − 10·73-s − 15·79-s − 15·83-s − 10·85-s + 9·89-s + 2·95-s − 8·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.38·13-s + 1.21·17-s − 0.229·19-s − 0.834·23-s − 1/5·25-s − 1.85·29-s − 1.79·31-s + 0.328·37-s + 1.52·43-s + 1.16·47-s − 49-s − 0.137·53-s − 0.650·59-s − 1.28·61-s + 1.24·65-s − 0.977·67-s − 0.356·71-s − 1.17·73-s − 1.68·79-s − 1.64·83-s − 1.08·85-s + 0.953·89-s + 0.205·95-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 331056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 331056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 15 T + p T^{2} \) |
| 89 | \( 1 - 9 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.71633508315292, −12.43586871621260, −11.89212482936191, −11.49592479994576, −11.15910033714825, −10.47996453273594, −10.16687794491391, −9.604682089474520, −9.132767249422701, −8.862824044020710, −7.912647740678857, −7.747489756676847, −7.335490062634772, −7.149716865810805, −6.146220593103489, −5.664087966053465, −5.511125342172552, −4.614325206422003, −4.258456646963592, −3.808530609033698, −3.189872967053188, −2.760828427154591, −1.897620238661741, −1.606638328711879, −0.5015549490229828, 0,
0.5015549490229828, 1.606638328711879, 1.897620238661741, 2.760828427154591, 3.189872967053188, 3.808530609033698, 4.258456646963592, 4.614325206422003, 5.511125342172552, 5.664087966053465, 6.146220593103489, 7.149716865810805, 7.335490062634772, 7.747489756676847, 7.912647740678857, 8.862824044020710, 9.132767249422701, 9.604682089474520, 10.16687794491391, 10.47996453273594, 11.15910033714825, 11.49592479994576, 11.89212482936191, 12.43586871621260, 12.71633508315292
