| L(s) = 1 | + 2.23·2-s + 3.00·4-s + 0.618·5-s − 7-s + 2.23·8-s + 1.38·10-s − 3.23·13-s − 2.23·14-s − 0.999·16-s + 4.85·17-s − 2.85·19-s + 1.85·20-s − 4.38·23-s − 4.61·25-s − 7.23·26-s − 3.00·28-s − 6·29-s − 3.09·31-s − 6.70·32-s + 10.8·34-s − 0.618·35-s + 4.61·37-s − 6.38·38-s + 1.38·40-s + 7.38·41-s − 9.70·43-s − 9.79·46-s + ⋯ |
| L(s) = 1 | + 1.58·2-s + 1.50·4-s + 0.276·5-s − 0.377·7-s + 0.790·8-s + 0.437·10-s − 0.897·13-s − 0.597·14-s − 0.249·16-s + 1.17·17-s − 0.654·19-s + 0.414·20-s − 0.913·23-s − 0.923·25-s − 1.41·26-s − 0.566·28-s − 1.11·29-s − 0.555·31-s − 1.18·32-s + 1.86·34-s − 0.104·35-s + 0.759·37-s − 1.03·38-s + 0.218·40-s + 1.15·41-s − 1.48·43-s − 1.44·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7623 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.23T + 2T^{2} \) |
| 5 | \( 1 - 0.618T + 5T^{2} \) |
| 13 | \( 1 + 3.23T + 13T^{2} \) |
| 17 | \( 1 - 4.85T + 17T^{2} \) |
| 19 | \( 1 + 2.85T + 19T^{2} \) |
| 23 | \( 1 + 4.38T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + 3.09T + 31T^{2} \) |
| 37 | \( 1 - 4.61T + 37T^{2} \) |
| 41 | \( 1 - 7.38T + 41T^{2} \) |
| 43 | \( 1 + 9.70T + 43T^{2} \) |
| 47 | \( 1 + 4.47T + 47T^{2} \) |
| 53 | \( 1 + 6.76T + 53T^{2} \) |
| 59 | \( 1 + 3.23T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 4.94T + 71T^{2} \) |
| 73 | \( 1 + 13.7T + 73T^{2} \) |
| 79 | \( 1 - 10T + 79T^{2} \) |
| 83 | \( 1 - 16.4T + 83T^{2} \) |
| 89 | \( 1 + 5.61T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−7.46099236116966791242931639341, −6.43991124984964430973287852262, −6.03298015405514428330868083256, −5.37633671875196178118874100905, −4.73057518256057387468968004787, −3.90632933207306953419099121276, −3.36998442567772693718500317289, −2.47209053735098214787032186700, −1.76488966538656580099674529915, 0,
1.76488966538656580099674529915, 2.47209053735098214787032186700, 3.36998442567772693718500317289, 3.90632933207306953419099121276, 4.73057518256057387468968004787, 5.37633671875196178118874100905, 6.03298015405514428330868083256, 6.43991124984964430973287852262, 7.46099236116966791242931639341
