| L(s) = 1 | + 2.61·2-s − 2.23·3-s + 4.85·4-s − 5.85·6-s − 4.23·7-s + 7.47·8-s + 2.00·9-s + 5.47·11-s − 10.8·12-s − 2·13-s − 11.0·14-s + 9.85·16-s − 4.85·17-s + 5.23·18-s + 9.47·21-s + 14.3·22-s − 1.23·23-s − 16.7·24-s − 5.23·26-s + 2.23·27-s − 20.5·28-s + 6·29-s − 1.85·31-s + 10.8·32-s − 12.2·33-s − 12.7·34-s + 9.70·36-s + ⋯ |
| L(s) = 1 | + 1.85·2-s − 1.29·3-s + 2.42·4-s − 2.38·6-s − 1.60·7-s + 2.64·8-s + 0.666·9-s + 1.64·11-s − 3.13·12-s − 0.554·13-s − 2.96·14-s + 2.46·16-s − 1.17·17-s + 1.23·18-s + 2.06·21-s + 3.05·22-s − 0.257·23-s − 3.41·24-s − 1.02·26-s + 0.430·27-s − 3.88·28-s + 1.11·29-s − 0.333·31-s + 1.91·32-s − 2.13·33-s − 2.17·34-s + 1.61·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.61T + 2T^{2} \) |
| 3 | \( 1 + 2.23T + 3T^{2} \) |
| 7 | \( 1 + 4.23T + 7T^{2} \) |
| 11 | \( 1 - 5.47T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 + 4.85T + 17T^{2} \) |
| 23 | \( 1 + 1.23T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 + 1.85T + 31T^{2} \) |
| 37 | \( 1 - 4.14T + 37T^{2} \) |
| 41 | \( 1 + 6.38T + 41T^{2} \) |
| 43 | \( 1 - 10.5T + 43T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 + 6.38T + 53T^{2} \) |
| 59 | \( 1 + 2.76T + 59T^{2} \) |
| 61 | \( 1 + 5.56T + 61T^{2} \) |
| 67 | \( 1 + 8.70T + 67T^{2} \) |
| 71 | \( 1 + 4.52T + 71T^{2} \) |
| 73 | \( 1 + 10.7T + 73T^{2} \) |
| 79 | \( 1 - T + 79T^{2} \) |
| 83 | \( 1 - 1.09T + 83T^{2} \) |
| 89 | \( 1 - 3.70T + 89T^{2} \) |
| 97 | \( 1 - 7.38T + 97T^{2} \) |
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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
−6.75321937530532642955377493325, −6.33755763286343462560276210655, −6.21949296521665716108503268361, −5.34307406978642790013979298324, −4.51234158441839076199605284279, −4.14663291823189629991153575313, −3.25817284211849500716738659378, −2.60175035412979946202994564452, −1.39573751911690929936733733983, 0,
1.39573751911690929936733733983, 2.60175035412979946202994564452, 3.25817284211849500716738659378, 4.14663291823189629991153575313, 4.51234158441839076199605284279, 5.34307406978642790013979298324, 6.21949296521665716108503268361, 6.33755763286343462560276210655, 6.75321937530532642955377493325
