| L(s) = 1 | + 2-s + 4-s + 0.592·7-s + 8-s + 11-s + 6.12·13-s + 0.592·14-s + 16-s + 1.40·17-s + 7.52·19-s + 22-s − 5.52·23-s + 6.12·26-s + 0.592·28-s − 1.59·29-s + 3.59·31-s + 32-s + 1.40·34-s + 0.592·37-s + 7.52·38-s − 9.64·41-s − 8.12·43-s + 44-s − 5.52·46-s + 4.59·47-s − 6.64·49-s + 6.12·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s + 0.224·7-s + 0.353·8-s + 0.301·11-s + 1.69·13-s + 0.158·14-s + 0.250·16-s + 0.341·17-s + 1.72·19-s + 0.213·22-s − 1.15·23-s + 1.20·26-s + 0.112·28-s − 0.295·29-s + 0.645·31-s + 0.176·32-s + 0.241·34-s + 0.0974·37-s + 1.22·38-s − 1.50·41-s − 1.23·43-s + 0.150·44-s − 0.815·46-s + 0.669·47-s − 0.949·49-s + 0.848·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.839838292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.839838292\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 0.592T + 7T^{2} \) |
| 13 | \( 1 - 6.12T + 13T^{2} \) |
| 17 | \( 1 - 1.40T + 17T^{2} \) |
| 19 | \( 1 - 7.52T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 + 1.59T + 29T^{2} \) |
| 31 | \( 1 - 3.59T + 31T^{2} \) |
| 37 | \( 1 - 0.592T + 37T^{2} \) |
| 41 | \( 1 + 9.64T + 41T^{2} \) |
| 43 | \( 1 + 8.12T + 43T^{2} \) |
| 47 | \( 1 - 4.59T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 4.59T + 59T^{2} \) |
| 61 | \( 1 - 5.18T + 61T^{2} \) |
| 67 | \( 1 - 2T + 67T^{2} \) |
| 71 | \( 1 - 8.34T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 - 9.40T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 - 7.81T + 97T^{2} \) |
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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
−8.259702404137302762222845694288, −7.43886959621838296273698671639, −6.70470772632545304608143243763, −5.92481933010629849069484649739, −5.43816849268121930830906595409, −4.51467967402088519734332088926, −3.64051331569269209950377751846, −3.20450353298246103671607540858, −1.89368541380107583966762264572, −1.04918822783148248317128353734,
1.04918822783148248317128353734, 1.89368541380107583966762264572, 3.20450353298246103671607540858, 3.64051331569269209950377751846, 4.51467967402088519734332088926, 5.43816849268121930830906595409, 5.92481933010629849069484649739, 6.70470772632545304608143243763, 7.43886959621838296273698671639, 8.259702404137302762222845694288
