| L(s) = 1 | + 0.823·2-s + 2.66·3-s − 1.32·4-s − 3.16·5-s + 2.19·6-s + 7-s − 2.73·8-s + 4.07·9-s − 2.60·10-s − 5.94·11-s − 3.51·12-s + 0.823·14-s − 8.41·15-s + 0.390·16-s − 2.69·17-s + 3.35·18-s − 1.95·19-s + 4.17·20-s + 2.66·21-s − 4.89·22-s − 2.72·23-s − 7.27·24-s + 4.99·25-s + 2.86·27-s − 1.32·28-s − 5.99·29-s − 6.92·30-s + ⋯ |
| L(s) = 1 | + 0.582·2-s + 1.53·3-s − 0.660·4-s − 1.41·5-s + 0.894·6-s + 0.377·7-s − 0.967·8-s + 1.35·9-s − 0.823·10-s − 1.79·11-s − 1.01·12-s + 0.220·14-s − 2.17·15-s + 0.0976·16-s − 0.654·17-s + 0.791·18-s − 0.448·19-s + 0.934·20-s + 0.580·21-s − 1.04·22-s − 0.569·23-s − 1.48·24-s + 0.999·25-s + 0.551·27-s − 0.249·28-s − 1.11·29-s − 1.26·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.823T + 2T^{2} \) |
| 3 | \( 1 - 2.66T + 3T^{2} \) |
| 5 | \( 1 + 3.16T + 5T^{2} \) |
| 11 | \( 1 + 5.94T + 11T^{2} \) |
| 17 | \( 1 + 2.69T + 17T^{2} \) |
| 19 | \( 1 + 1.95T + 19T^{2} \) |
| 23 | \( 1 + 2.72T + 23T^{2} \) |
| 29 | \( 1 + 5.99T + 29T^{2} \) |
| 31 | \( 1 + 1.15T + 31T^{2} \) |
| 37 | \( 1 - 6.50T + 37T^{2} \) |
| 41 | \( 1 + 3.73T + 41T^{2} \) |
| 43 | \( 1 - 6.99T + 43T^{2} \) |
| 47 | \( 1 + 0.456T + 47T^{2} \) |
| 53 | \( 1 - 0.399T + 53T^{2} \) |
| 59 | \( 1 - 4.80T + 59T^{2} \) |
| 61 | \( 1 + 1.15T + 61T^{2} \) |
| 67 | \( 1 + 6.27T + 67T^{2} \) |
| 71 | \( 1 + 4.50T + 71T^{2} \) |
| 73 | \( 1 + 8.30T + 73T^{2} \) |
| 79 | \( 1 + 7.91T + 79T^{2} \) |
| 83 | \( 1 - 6.19T + 83T^{2} \) |
| 89 | \( 1 + 3.56T + 89T^{2} \) |
| 97 | \( 1 - 3.42T + 97T^{2} \) |
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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
−9.049574181825038612782490319646, −8.423366764508955456872279525137, −7.86673615642151917866738646691, −7.35453281153237027123278790974, −5.73849633065175257049015624323, −4.62664865461162016549144421349, −4.05861267221818581646972542170, −3.20961900056412451607801728881, −2.34308138255150873017426840969, 0,
2.34308138255150873017426840969, 3.20961900056412451607801728881, 4.05861267221818581646972542170, 4.62664865461162016549144421349, 5.73849633065175257049015624323, 7.35453281153237027123278790974, 7.86673615642151917866738646691, 8.423366764508955456872279525137, 9.049574181825038612782490319646
