L(s) = 1 | + 2-s − 2.30·3-s + 4-s − 2.85·5-s − 2.30·6-s + 7-s + 8-s + 2.31·9-s − 2.85·10-s − 1.11·11-s − 2.30·12-s + 2.44·13-s + 14-s + 6.57·15-s + 16-s + 1.74·17-s + 2.31·18-s − 0.0622·19-s − 2.85·20-s − 2.30·21-s − 1.11·22-s − 4.67·23-s − 2.30·24-s + 3.13·25-s + 2.44·26-s + 1.57·27-s + 28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.33·3-s + 0.5·4-s − 1.27·5-s − 0.941·6-s + 0.377·7-s + 0.353·8-s + 0.772·9-s − 0.902·10-s − 0.336·11-s − 0.665·12-s + 0.679·13-s + 0.267·14-s + 1.69·15-s + 0.250·16-s + 0.422·17-s + 0.546·18-s − 0.0142·19-s − 0.637·20-s − 0.503·21-s − 0.238·22-s − 0.974·23-s − 0.470·24-s + 0.627·25-s + 0.480·26-s + 0.302·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 431 | \( 1 + T \) |
good | 3 | \( 1 + 2.30T + 3T^{2} \) |
| 5 | \( 1 + 2.85T + 5T^{2} \) |
| 11 | \( 1 + 1.11T + 11T^{2} \) |
| 13 | \( 1 - 2.44T + 13T^{2} \) |
| 17 | \( 1 - 1.74T + 17T^{2} \) |
| 19 | \( 1 + 0.0622T + 19T^{2} \) |
| 23 | \( 1 + 4.67T + 23T^{2} \) |
| 29 | \( 1 + 9.83T + 29T^{2} \) |
| 31 | \( 1 + 0.366T + 31T^{2} \) |
| 37 | \( 1 + 1.57T + 37T^{2} \) |
| 41 | \( 1 - 11.0T + 41T^{2} \) |
| 43 | \( 1 - 1.33T + 43T^{2} \) |
| 47 | \( 1 - 10.0T + 47T^{2} \) |
| 53 | \( 1 - 3.40T + 53T^{2} \) |
| 59 | \( 1 - 6.27T + 59T^{2} \) |
| 61 | \( 1 - 2.39T + 61T^{2} \) |
| 67 | \( 1 + 13.7T + 67T^{2} \) |
| 71 | \( 1 - 6.35T + 71T^{2} \) |
| 73 | \( 1 - 3.56T + 73T^{2} \) |
| 79 | \( 1 - 8.90T + 79T^{2} \) |
| 83 | \( 1 + 10.5T + 83T^{2} \) |
| 89 | \( 1 + 6.43T + 89T^{2} \) |
| 97 | \( 1 + 0.643T + 97T^{2} \) |
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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
−7.54414860052682142315478164130, −7.01412266847184265202089711941, −5.88780640204769767780795182378, −5.74974462806835990837227971185, −4.84665560701640408378332423208, −4.05865769024692710972488094700, −3.67145333087341656666346817176, −2.40326998126095232235555006964, −1.10923141889783700172648808879, 0,
1.10923141889783700172648808879, 2.40326998126095232235555006964, 3.67145333087341656666346817176, 4.05865769024692710972488094700, 4.84665560701640408378332423208, 5.74974462806835990837227971185, 5.88780640204769767780795182378, 7.01412266847184265202089711941, 7.54414860052682142315478164130