| L(s) = 1 | − 2-s + 1.41·3-s + 4-s − 1.43·5-s − 1.41·6-s + 4.06·7-s − 8-s − 0.997·9-s + 1.43·10-s − 1.37·11-s + 1.41·12-s − 5.36·13-s − 4.06·14-s − 2.03·15-s + 16-s + 6.61·17-s + 0.997·18-s + 5.12·19-s − 1.43·20-s + 5.74·21-s + 1.37·22-s − 5.83·23-s − 1.41·24-s − 2.92·25-s + 5.36·26-s − 5.65·27-s + 4.06·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.817·3-s + 0.5·4-s − 0.643·5-s − 0.577·6-s + 1.53·7-s − 0.353·8-s − 0.332·9-s + 0.455·10-s − 0.415·11-s + 0.408·12-s − 1.48·13-s − 1.08·14-s − 0.525·15-s + 0.250·16-s + 1.60·17-s + 0.235·18-s + 1.17·19-s − 0.321·20-s + 1.25·21-s + 0.294·22-s − 1.21·23-s − 0.288·24-s − 0.585·25-s + 1.05·26-s − 1.08·27-s + 0.767·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 5 | \( 1 + 1.43T + 5T^{2} \) |
| 7 | \( 1 - 4.06T + 7T^{2} \) |
| 11 | \( 1 + 1.37T + 11T^{2} \) |
| 13 | \( 1 + 5.36T + 13T^{2} \) |
| 17 | \( 1 - 6.61T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 + 1.70T + 29T^{2} \) |
| 31 | \( 1 + 9.67T + 31T^{2} \) |
| 37 | \( 1 + 9.40T + 37T^{2} \) |
| 41 | \( 1 - 5.15T + 41T^{2} \) |
| 43 | \( 1 + 2.89T + 43T^{2} \) |
| 47 | \( 1 - 2.48T + 47T^{2} \) |
| 53 | \( 1 - 5.70T + 53T^{2} \) |
| 59 | \( 1 - 5.23T + 59T^{2} \) |
| 61 | \( 1 + 13.0T + 61T^{2} \) |
| 67 | \( 1 - 13.0T + 67T^{2} \) |
| 71 | \( 1 + 8.86T + 71T^{2} \) |
| 73 | \( 1 + 1.48T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 1.36T + 83T^{2} \) |
| 89 | \( 1 - 1.96T + 89T^{2} \) |
| 97 | \( 1 + 9.47T + 97T^{2} \) |
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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
−7.903163713726520316238079553360, −7.66673662953051340710648411454, −7.25415517762742971925815803942, −5.46168192380678166368292710351, −5.40292953803145368378540645212, −4.07760704850399572212526464144, −3.27083573914119024802846546045, −2.31989727171529447204296414909, −1.54344373048327229825575775484, 0,
1.54344373048327229825575775484, 2.31989727171529447204296414909, 3.27083573914119024802846546045, 4.07760704850399572212526464144, 5.40292953803145368378540645212, 5.46168192380678166368292710351, 7.25415517762742971925815803942, 7.66673662953051340710648411454, 7.903163713726520316238079553360
