| L(s) = 1 | − 3-s + 2·5-s − 7-s − 2·9-s + 5·11-s − 2·15-s + 2·17-s − 4·19-s + 21-s + 9·23-s − 25-s + 5·27-s − 5·31-s − 5·33-s − 2·35-s − 3·37-s + 5·41-s − 4·43-s − 4·45-s − 13·47-s + 49-s − 2·51-s − 14·53-s + 10·55-s + 4·57-s − 6·59-s + 13·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.377·7-s − 2/3·9-s + 1.50·11-s − 0.516·15-s + 0.485·17-s − 0.917·19-s + 0.218·21-s + 1.87·23-s − 1/5·25-s + 0.962·27-s − 0.898·31-s − 0.870·33-s − 0.338·35-s − 0.493·37-s + 0.780·41-s − 0.609·43-s − 0.596·45-s − 1.89·47-s + 1/7·49-s − 0.280·51-s − 1.92·53-s + 1.34·55-s + 0.529·57-s − 0.781·59-s + 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−14.22720273072778, −13.98697750018614, −13.10064633486922, −12.90790049494996, −12.34407041041046, −11.74928539072646, −11.23562694264072, −10.95840103779811, −10.31388767571312, −9.631671618441684, −9.362692689931787, −8.839182987912091, −8.360718086703210, −7.584809169588520, −6.730785964995556, −6.602077596269834, −6.055738400942083, −5.527601561457505, −4.961549258312725, −4.416690502872191, −3.425025377448213, −3.243054347317550, −2.251297750640166, −1.616814476277018, −0.9509638853549972, 0,
0.9509638853549972, 1.616814476277018, 2.251297750640166, 3.243054347317550, 3.425025377448213, 4.416690502872191, 4.961549258312725, 5.527601561457505, 6.055738400942083, 6.602077596269834, 6.730785964995556, 7.584809169588520, 8.360718086703210, 8.839182987912091, 9.362692689931787, 9.631671618441684, 10.31388767571312, 10.95840103779811, 11.23562694264072, 11.74928539072646, 12.34407041041046, 12.90790049494996, 13.10064633486922, 13.98697750018614, 14.22720273072778
