| L(s) = 1 | − 2-s + 4-s + 0.974·5-s + 7-s − 8-s − 0.974·10-s − 1.02·11-s − 0.655·13-s − 14-s + 16-s − 0.655·17-s + 0.319·19-s + 0.974·20-s + 1.02·22-s − 23-s − 4.04·25-s + 0.655·26-s + 28-s − 2·29-s + 9.70·31-s − 32-s + 0.655·34-s + 0.974·35-s + 11.3·37-s − 0.319·38-s − 0.974·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.435·5-s + 0.377·7-s − 0.353·8-s − 0.308·10-s − 0.309·11-s − 0.181·13-s − 0.267·14-s + 0.250·16-s − 0.158·17-s + 0.0732·19-s + 0.217·20-s + 0.218·22-s − 0.208·23-s − 0.809·25-s + 0.128·26-s + 0.188·28-s − 0.371·29-s + 1.74·31-s − 0.176·32-s + 0.112·34-s + 0.164·35-s + 1.86·37-s − 0.0518·38-s − 0.154·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.417012691\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.417012691\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 0.974T + 5T^{2} \) |
| 11 | \( 1 + 1.02T + 11T^{2} \) |
| 13 | \( 1 + 0.655T + 13T^{2} \) |
| 17 | \( 1 + 0.655T + 17T^{2} \) |
| 19 | \( 1 - 0.319T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 9.70T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 - 4.28T + 43T^{2} \) |
| 47 | \( 1 + 5.70T + 47T^{2} \) |
| 53 | \( 1 - 7.33T + 53T^{2} \) |
| 59 | \( 1 - 2.05T + 59T^{2} \) |
| 61 | \( 1 - 2.92T + 61T^{2} \) |
| 67 | \( 1 - 7.61T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 6.36T + 73T^{2} \) |
| 79 | \( 1 + 2.36T + 79T^{2} \) |
| 83 | \( 1 - 15.7T + 83T^{2} \) |
| 89 | \( 1 - 18.0T + 89T^{2} \) |
| 97 | \( 1 - 16.0T + 97T^{2} \) |
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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
−8.782279200148018712801513912038, −7.968937193995462778238996977853, −7.53105712704095768681225784959, −6.46809848800046487246933990072, −5.89961278416583331735461010455, −4.95143751698466238212976506455, −4.03454059155895161897663429468, −2.77456126399682010053581718546, −2.03403062681527249207984687326, −0.817854113836109809479231920072,
0.817854113836109809479231920072, 2.03403062681527249207984687326, 2.77456126399682010053581718546, 4.03454059155895161897663429468, 4.95143751698466238212976506455, 5.89961278416583331735461010455, 6.46809848800046487246933990072, 7.53105712704095768681225784959, 7.968937193995462778238996977853, 8.782279200148018712801513912038
