| L(s) = 1 | + 0.494·2-s − 1.75·4-s + 3.82·5-s − 3.40·7-s − 1.85·8-s + 1.88·10-s + 5.21·11-s + 3.85·13-s − 1.68·14-s + 2.59·16-s + 3.52·17-s − 19-s − 6.71·20-s + 2.57·22-s + 3.25·23-s + 9.61·25-s + 1.90·26-s + 5.98·28-s − 1.82·29-s + 5.41·31-s + 4.99·32-s + 1.74·34-s − 13.0·35-s − 8.34·37-s − 0.494·38-s − 7.09·40-s + 8.29·41-s + ⋯ |
| L(s) = 1 | + 0.349·2-s − 0.877·4-s + 1.70·5-s − 1.28·7-s − 0.656·8-s + 0.597·10-s + 1.57·11-s + 1.06·13-s − 0.450·14-s + 0.648·16-s + 0.855·17-s − 0.229·19-s − 1.50·20-s + 0.549·22-s + 0.677·23-s + 1.92·25-s + 0.373·26-s + 1.13·28-s − 0.338·29-s + 0.971·31-s + 0.882·32-s + 0.298·34-s − 2.20·35-s − 1.37·37-s − 0.0801·38-s − 1.12·40-s + 1.29·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8037 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.952799815\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.952799815\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + T \) |
| 47 | \( 1 + T \) |
| good | 2 | \( 1 - 0.494T + 2T^{2} \) |
| 5 | \( 1 - 3.82T + 5T^{2} \) |
| 7 | \( 1 + 3.40T + 7T^{2} \) |
| 11 | \( 1 - 5.21T + 11T^{2} \) |
| 13 | \( 1 - 3.85T + 13T^{2} \) |
| 17 | \( 1 - 3.52T + 17T^{2} \) |
| 23 | \( 1 - 3.25T + 23T^{2} \) |
| 29 | \( 1 + 1.82T + 29T^{2} \) |
| 31 | \( 1 - 5.41T + 31T^{2} \) |
| 37 | \( 1 + 8.34T + 37T^{2} \) |
| 41 | \( 1 - 8.29T + 41T^{2} \) |
| 43 | \( 1 - 3.89T + 43T^{2} \) |
| 53 | \( 1 + 5.56T + 53T^{2} \) |
| 59 | \( 1 + 7.79T + 59T^{2} \) |
| 61 | \( 1 + 13.1T + 61T^{2} \) |
| 67 | \( 1 + 3.15T + 67T^{2} \) |
| 71 | \( 1 - 7.73T + 71T^{2} \) |
| 73 | \( 1 + 10.8T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 - 10.6T + 83T^{2} \) |
| 89 | \( 1 - 3.54T + 89T^{2} \) |
| 97 | \( 1 - 12.4T + 97T^{2} \) |
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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
−7.892590219164660481924845632680, −6.67117182516712459583611188669, −6.30384544029307465775464328610, −5.90731206816310340849239681105, −5.16081076872618107359179059753, −4.24265922550482133801517265167, −3.45638462792895033927894165263, −2.94064450853549227580927746408, −1.62909888256047007306252922895, −0.881356499691484736839267495383,
0.881356499691484736839267495383, 1.62909888256047007306252922895, 2.94064450853549227580927746408, 3.45638462792895033927894165263, 4.24265922550482133801517265167, 5.16081076872618107359179059753, 5.90731206816310340849239681105, 6.30384544029307465775464328610, 6.67117182516712459583611188669, 7.892590219164660481924845632680
