| L(s) = 1 | + 1.95·2-s − 1.67·3-s + 1.83·4-s + 2.80·5-s − 3.27·6-s + 3.08·7-s − 0.315·8-s − 0.209·9-s + 5.50·10-s − 1.74·11-s − 3.07·12-s − 2.73·13-s + 6.04·14-s − 4.69·15-s − 4.29·16-s − 0.616·17-s − 0.409·18-s − 5.70·19-s + 5.16·20-s − 5.15·21-s − 3.41·22-s − 4.59·23-s + 0.527·24-s + 2.89·25-s − 5.35·26-s + 5.36·27-s + 5.67·28-s + ⋯ |
| L(s) = 1 | + 1.38·2-s − 0.964·3-s + 0.919·4-s + 1.25·5-s − 1.33·6-s + 1.16·7-s − 0.111·8-s − 0.0697·9-s + 1.74·10-s − 0.526·11-s − 0.886·12-s − 0.757·13-s + 1.61·14-s − 1.21·15-s − 1.07·16-s − 0.149·17-s − 0.0965·18-s − 1.30·19-s + 1.15·20-s − 1.12·21-s − 0.729·22-s − 0.957·23-s + 0.107·24-s + 0.578·25-s − 1.05·26-s + 1.03·27-s + 1.07·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 37 | \( 1 + T \) |
| 109 | \( 1 + T \) |
| good | 2 | \( 1 - 1.95T + 2T^{2} \) |
| 3 | \( 1 + 1.67T + 3T^{2} \) |
| 5 | \( 1 - 2.80T + 5T^{2} \) |
| 7 | \( 1 - 3.08T + 7T^{2} \) |
| 11 | \( 1 + 1.74T + 11T^{2} \) |
| 13 | \( 1 + 2.73T + 13T^{2} \) |
| 17 | \( 1 + 0.616T + 17T^{2} \) |
| 19 | \( 1 + 5.70T + 19T^{2} \) |
| 23 | \( 1 + 4.59T + 23T^{2} \) |
| 29 | \( 1 + 7.61T + 29T^{2} \) |
| 31 | \( 1 + 9.29T + 31T^{2} \) |
| 41 | \( 1 + 2.53T + 41T^{2} \) |
| 43 | \( 1 + 4.26T + 43T^{2} \) |
| 47 | \( 1 - 4.53T + 47T^{2} \) |
| 53 | \( 1 - 4.14T + 53T^{2} \) |
| 59 | \( 1 + 11.7T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 - 7.61T + 67T^{2} \) |
| 71 | \( 1 - 13.4T + 71T^{2} \) |
| 73 | \( 1 - 3.33T + 73T^{2} \) |
| 79 | \( 1 + 3.75T + 79T^{2} \) |
| 83 | \( 1 + 2.28T + 83T^{2} \) |
| 89 | \( 1 - 11.6T + 89T^{2} \) |
| 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
−7.937037886986302809151334373460, −6.92797594989147744102080614283, −6.22544271213481751710807944747, −5.48753288580239044445248437336, −5.31977170554557516302645937919, −4.58779033070357417871690122116, −3.68318154633881390530853121262, −2.28568038456954770664218300345, −1.95882249097284784809118652354, 0,
1.95882249097284784809118652354, 2.28568038456954770664218300345, 3.68318154633881390530853121262, 4.58779033070357417871690122116, 5.31977170554557516302645937919, 5.48753288580239044445248437336, 6.22544271213481751710807944747, 6.92797594989147744102080614283, 7.937037886986302809151334373460
