| L(s) = 1 | − 2-s + 0.381·3-s + 4-s − 4.23·5-s − 0.381·6-s − 7-s − 8-s − 2.85·9-s + 4.23·10-s + 3·11-s + 0.381·12-s − 3.85·13-s + 14-s − 1.61·15-s + 16-s + 1.85·17-s + 2.85·18-s − 4.23·20-s − 0.381·21-s − 3·22-s − 6.70·23-s − 0.381·24-s + 12.9·25-s + 3.85·26-s − 2.23·27-s − 28-s − 8.23·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.220·3-s + 0.5·4-s − 1.89·5-s − 0.155·6-s − 0.377·7-s − 0.353·8-s − 0.951·9-s + 1.33·10-s + 0.904·11-s + 0.110·12-s − 1.06·13-s + 0.267·14-s − 0.417·15-s + 0.250·16-s + 0.449·17-s + 0.672·18-s − 0.947·20-s − 0.0833·21-s − 0.639·22-s − 1.39·23-s − 0.0779·24-s + 2.58·25-s + 0.755·26-s − 0.430·27-s − 0.188·28-s − 1.52·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.2185175749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2185175749\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.381T + 3T^{2} \) |
| 5 | \( 1 + 4.23T + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 3.85T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 23 | \( 1 + 6.70T + 23T^{2} \) |
| 29 | \( 1 + 8.23T + 29T^{2} \) |
| 31 | \( 1 - 5T + 31T^{2} \) |
| 37 | \( 1 + 5T + 37T^{2} \) |
| 41 | \( 1 + 4.09T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 6.23T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 10.2T + 59T^{2} \) |
| 61 | \( 1 + 6.70T + 61T^{2} \) |
| 67 | \( 1 + 4.56T + 67T^{2} \) |
| 71 | \( 1 - 3.52T + 71T^{2} \) |
| 73 | \( 1 + 9.09T + 73T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + 6.47T + 83T^{2} \) |
| 89 | \( 1 + 1.38T + 89T^{2} \) |
| 97 | \( 1 + 5.29T + 97T^{2} \) |
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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
−8.191891778338104501150534830947, −7.72758387381972566852892755863, −7.01120316441412238598491720538, −6.36794444436543299655366913367, −5.30194560903042443945436817170, −4.35823881154476914691464121369, −3.50298659509189696801340096952, −3.07755957027637096993872645079, −1.79116867772539370725798303792, −0.26849653969629374787576615987,
0.26849653969629374787576615987, 1.79116867772539370725798303792, 3.07755957027637096993872645079, 3.50298659509189696801340096952, 4.35823881154476914691464121369, 5.30194560903042443945436817170, 6.36794444436543299655366913367, 7.01120316441412238598491720538, 7.72758387381972566852892755863, 8.191891778338104501150534830947
