| L(s) = 1 | + 3-s − 2·4-s + 2·5-s − 7-s + 9-s + 11-s − 2·12-s − 13-s + 2·15-s + 4·16-s + 2·17-s − 4·20-s − 21-s − 23-s − 25-s + 27-s + 2·28-s + 8·29-s − 8·31-s + 33-s − 2·35-s − 2·36-s − 7·37-s − 39-s − 3·41-s − 2·44-s + 2·45-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 4-s + 0.894·5-s − 0.377·7-s + 1/3·9-s + 0.301·11-s − 0.577·12-s − 0.277·13-s + 0.516·15-s + 16-s + 0.485·17-s − 0.894·20-s − 0.218·21-s − 0.208·23-s − 1/5·25-s + 0.192·27-s + 0.377·28-s + 1.48·29-s − 1.43·31-s + 0.174·33-s − 0.338·35-s − 1/3·36-s − 1.15·37-s − 0.160·39-s − 0.468·41-s − 0.301·44-s + 0.298·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 83391 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 7 T + p T^{2} \) |
| 71 | \( 1 - 11 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−14.01851218670338, −13.83282389991879, −13.31172012397372, −12.83332384459858, −12.32115444064474, −11.99938173955872, −11.11774539786247, −10.42932579804496, −10.06928398233394, −9.659854599640240, −9.185860759670997, −8.841239856341987, −8.101981988051323, −7.877076881081837, −6.906542093540874, −6.627162454902292, −5.841809909784067, −5.289780158035872, −4.964221928182219, −4.075423021152096, −3.672244469888724, −3.083144792127076, −2.325314555172665, −1.677658907483293, −0.9587887481045757, 0,
0.9587887481045757, 1.677658907483293, 2.325314555172665, 3.083144792127076, 3.672244469888724, 4.075423021152096, 4.964221928182219, 5.289780158035872, 5.841809909784067, 6.627162454902292, 6.906542093540874, 7.877076881081837, 8.101981988051323, 8.841239856341987, 9.185860759670997, 9.659854599640240, 10.06928398233394, 10.42932579804496, 11.11774539786247, 11.99938173955872, 12.32115444064474, 12.83332384459858, 13.31172012397372, 13.83282389991879, 14.01851218670338
