| L(s) = 1 | + 2-s + 2.64·3-s + 4-s + 0.670·5-s + 2.64·6-s + 3.14·7-s + 8-s + 3.97·9-s + 0.670·10-s + 0.295·11-s + 2.64·12-s − 0.535·13-s + 3.14·14-s + 1.77·15-s + 16-s − 4.36·17-s + 3.97·18-s + 1.27·19-s + 0.670·20-s + 8.29·21-s + 0.295·22-s + 6.35·23-s + 2.64·24-s − 4.55·25-s − 0.535·26-s + 2.56·27-s + 3.14·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.52·3-s + 0.5·4-s + 0.299·5-s + 1.07·6-s + 1.18·7-s + 0.353·8-s + 1.32·9-s + 0.212·10-s + 0.0889·11-s + 0.762·12-s − 0.148·13-s + 0.839·14-s + 0.457·15-s + 0.250·16-s − 1.05·17-s + 0.936·18-s + 0.291·19-s + 0.149·20-s + 1.81·21-s + 0.0628·22-s + 1.32·23-s + 0.539·24-s − 0.910·25-s − 0.105·26-s + 0.494·27-s + 0.593·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4022 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.438268130\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.438268130\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 2011 | \( 1 + T \) |
| good | 3 | \( 1 - 2.64T + 3T^{2} \) |
| 5 | \( 1 - 0.670T + 5T^{2} \) |
| 7 | \( 1 - 3.14T + 7T^{2} \) |
| 11 | \( 1 - 0.295T + 11T^{2} \) |
| 13 | \( 1 + 0.535T + 13T^{2} \) |
| 17 | \( 1 + 4.36T + 17T^{2} \) |
| 19 | \( 1 - 1.27T + 19T^{2} \) |
| 23 | \( 1 - 6.35T + 23T^{2} \) |
| 29 | \( 1 + 0.0105T + 29T^{2} \) |
| 31 | \( 1 - 9.60T + 31T^{2} \) |
| 37 | \( 1 + 1.60T + 37T^{2} \) |
| 41 | \( 1 + 5.00T + 41T^{2} \) |
| 43 | \( 1 + 9.70T + 43T^{2} \) |
| 47 | \( 1 + 4.01T + 47T^{2} \) |
| 53 | \( 1 - 1.44T + 53T^{2} \) |
| 59 | \( 1 + 7.63T + 59T^{2} \) |
| 61 | \( 1 - 15.2T + 61T^{2} \) |
| 67 | \( 1 + 2.53T + 67T^{2} \) |
| 71 | \( 1 - 3.23T + 71T^{2} \) |
| 73 | \( 1 + 7.53T + 73T^{2} \) |
| 79 | \( 1 - 4.00T + 79T^{2} \) |
| 83 | \( 1 + 11.9T + 83T^{2} \) |
| 89 | \( 1 - 8.11T + 89T^{2} \) |
| 97 | \( 1 - 11.4T + 97T^{2} \) |
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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
−8.425562139725111056492386512305, −7.81774481864918165129132541819, −7.06427180170359240355913449182, −6.33035792442934307886290151368, −5.12011217467940093514492575522, −4.66736388456545832273686590073, −3.76935298365016386132696729032, −2.93247783898935670987107266074, −2.18108926164608695802514686680, −1.44547394773783424851335094274,
1.44547394773783424851335094274, 2.18108926164608695802514686680, 2.93247783898935670987107266074, 3.76935298365016386132696729032, 4.66736388456545832273686590073, 5.12011217467940093514492575522, 6.33035792442934307886290151368, 7.06427180170359240355913449182, 7.81774481864918165129132541819, 8.425562139725111056492386512305
