| L(s) = 1 | − 2·4-s − 3·5-s − 7-s − 2·11-s + 4·16-s + 5·17-s − 4·19-s + 6·20-s − 6·23-s + 4·25-s + 2·28-s − 29-s + 31-s + 3·35-s − 37-s − 2·41-s − 8·43-s + 4·44-s − 2·47-s + 49-s + 6·53-s + 6·55-s + 7·59-s − 8·64-s + 2·67-s − 10·68-s + 14·73-s + ⋯ |
| L(s) = 1 | − 4-s − 1.34·5-s − 0.377·7-s − 0.603·11-s + 16-s + 1.21·17-s − 0.917·19-s + 1.34·20-s − 1.25·23-s + 4/5·25-s + 0.377·28-s − 0.185·29-s + 0.179·31-s + 0.507·35-s − 0.164·37-s − 0.312·41-s − 1.21·43-s + 0.603·44-s − 0.291·47-s + 1/7·49-s + 0.824·53-s + 0.809·55-s + 0.911·59-s − 64-s + 0.244·67-s − 1.21·68-s + 1.63·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308763 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308763 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 7 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−12.98256571072103, −12.70030140282282, −12.16253209208521, −11.91121077641088, −11.39142676955619, −10.76755042273537, −10.31247942324523, −9.873653382603998, −9.611361519939672, −8.826464238328432, −8.294956594639685, −8.192801012270440, −7.759739562896422, −7.136503919138801, −6.686945959713949, −5.970185348430113, −5.466380669160357, −5.087075946650826, −4.407432423862122, −3.941559311020812, −3.705148284851786, −3.103584213724549, −2.499388809439340, −1.630721755169635, −0.8961557234207941, 0, 0,
0.8961557234207941, 1.630721755169635, 2.499388809439340, 3.103584213724549, 3.705148284851786, 3.941559311020812, 4.407432423862122, 5.087075946650826, 5.466380669160357, 5.970185348430113, 6.686945959713949, 7.136503919138801, 7.759739562896422, 8.192801012270440, 8.294956594639685, 8.826464238328432, 9.611361519939672, 9.873653382603998, 10.31247942324523, 10.76755042273537, 11.39142676955619, 11.91121077641088, 12.16253209208521, 12.70030140282282, 12.98256571072103
