| L(s) = 1 | − 0.531·2-s − 1.71·4-s − 2.93·7-s + 1.97·8-s + 5.92·11-s + 1.35·13-s + 1.56·14-s + 2.38·16-s + 6.16·17-s + 5.46·19-s − 3.15·22-s + 1.87·23-s − 0.717·26-s + 5.04·28-s + 7.96·29-s + 0.229·31-s − 5.22·32-s − 3.27·34-s − 37-s − 2.90·38-s + 3.07·41-s + 8.13·43-s − 10.1·44-s − 0.995·46-s − 9.06·47-s + 1.64·49-s − 2.31·52-s + ⋯ |
| L(s) = 1 | − 0.375·2-s − 0.858·4-s − 1.11·7-s + 0.698·8-s + 1.78·11-s + 0.374·13-s + 0.417·14-s + 0.596·16-s + 1.49·17-s + 1.25·19-s − 0.671·22-s + 0.390·23-s − 0.140·26-s + 0.953·28-s + 1.47·29-s + 0.0412·31-s − 0.922·32-s − 0.562·34-s − 0.164·37-s − 0.471·38-s + 0.480·41-s + 1.24·43-s − 1.53·44-s − 0.146·46-s − 1.32·47-s + 0.234·49-s − 0.321·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.683298299\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.683298299\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 37 | \( 1 + T \) |
| good | 2 | \( 1 + 0.531T + 2T^{2} \) |
| 7 | \( 1 + 2.93T + 7T^{2} \) |
| 11 | \( 1 - 5.92T + 11T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 6.16T + 17T^{2} \) |
| 19 | \( 1 - 5.46T + 19T^{2} \) |
| 23 | \( 1 - 1.87T + 23T^{2} \) |
| 29 | \( 1 - 7.96T + 29T^{2} \) |
| 31 | \( 1 - 0.229T + 31T^{2} \) |
| 41 | \( 1 - 3.07T + 41T^{2} \) |
| 43 | \( 1 - 8.13T + 43T^{2} \) |
| 47 | \( 1 + 9.06T + 47T^{2} \) |
| 53 | \( 1 - 2.63T + 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 4.90T + 61T^{2} \) |
| 67 | \( 1 - 4.03T + 67T^{2} \) |
| 71 | \( 1 - 9.03T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 + 5.53T + 79T^{2} \) |
| 83 | \( 1 - 1.13T + 83T^{2} \) |
| 89 | \( 1 + 8.14T + 89T^{2} \) |
| 97 | \( 1 + 17.9T + 97T^{2} \) |
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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
−7.973955467664704308006939496434, −7.01518694862388592812590008143, −6.57523293530340151452069056410, −5.70805218794598729287482901091, −5.09610987011027209604212856318, −4.01105588690945750865985198089, −3.63989309322251655798722940741, −2.86795046494324904497889465759, −1.25814097475873766987355162032, −0.836851591011418800736857589702,
0.836851591011418800736857589702, 1.25814097475873766987355162032, 2.86795046494324904497889465759, 3.63989309322251655798722940741, 4.01105588690945750865985198089, 5.09610987011027209604212856318, 5.70805218794598729287482901091, 6.57523293530340151452069056410, 7.01518694862388592812590008143, 7.973955467664704308006939496434
