| L(s) = 1 | − 2·3-s − 2·5-s + 2·7-s + 9-s + 4·11-s + 13-s + 4·15-s + 6·17-s − 4·21-s + 4·23-s − 25-s + 4·27-s + 4·31-s − 8·33-s − 4·35-s − 2·37-s − 2·39-s − 6·41-s + 2·43-s − 2·45-s + 12·47-s − 3·49-s − 12·51-s − 6·53-s − 8·55-s + 6·59-s + 10·61-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.894·5-s + 0.755·7-s + 1/3·9-s + 1.20·11-s + 0.277·13-s + 1.03·15-s + 1.45·17-s − 0.872·21-s + 0.834·23-s − 1/5·25-s + 0.769·27-s + 0.718·31-s − 1.39·33-s − 0.676·35-s − 0.328·37-s − 0.320·39-s − 0.937·41-s + 0.304·43-s − 0.298·45-s + 1.75·47-s − 3/7·49-s − 1.68·51-s − 0.824·53-s − 1.07·55-s + 0.781·59-s + 1.28·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 87464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 87464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| 29 | \( 1 \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−14.09318065473067, −13.85410344455174, −12.95652619858849, −12.41148478459791, −12.02219412697904, −11.57735232133638, −11.43090848699872, −10.87815867018413, −10.21931126116274, −9.885669430220584, −9.004587542350469, −8.579996096226251, −8.137863278738853, −7.294437157509871, −7.222289630202758, −6.361615603228443, −5.897227822704398, −5.438065850591433, −4.704166974139924, −4.442637298660463, −3.576474311028685, −3.289485085448742, −2.233351821496927, −1.168328719084913, −1.042615693800698, 0,
1.042615693800698, 1.168328719084913, 2.233351821496927, 3.289485085448742, 3.576474311028685, 4.442637298660463, 4.704166974139924, 5.438065850591433, 5.897227822704398, 6.361615603228443, 7.222289630202758, 7.294437157509871, 8.137863278738853, 8.579996096226251, 9.004587542350469, 9.885669430220584, 10.21931126116274, 10.87815867018413, 11.43090848699872, 11.57735232133638, 12.02219412697904, 12.41148478459791, 12.95652619858849, 13.85410344455174, 14.09318065473067
