| L(s) = 1 | + 2.04·3-s − 4.02·5-s + 1.43·7-s + 1.16·9-s − 4.74·11-s + 2.52·13-s − 8.22·15-s + 2.51·17-s + 3.36·19-s + 2.92·21-s + 2.10·23-s + 11.2·25-s − 3.73·27-s + 3.73·29-s − 9.98·31-s − 9.68·33-s − 5.77·35-s + 11.0·37-s + 5.15·39-s − 11.3·41-s − 2.86·43-s − 4.70·45-s + 5.89·47-s − 4.94·49-s + 5.12·51-s − 10.3·53-s + 19.1·55-s + ⋯ |
| L(s) = 1 | + 1.17·3-s − 1.80·5-s + 0.541·7-s + 0.389·9-s − 1.43·11-s + 0.700·13-s − 2.12·15-s + 0.609·17-s + 0.772·19-s + 0.638·21-s + 0.439·23-s + 2.24·25-s − 0.719·27-s + 0.693·29-s − 1.79·31-s − 1.68·33-s − 0.975·35-s + 1.81·37-s + 0.825·39-s − 1.77·41-s − 0.437·43-s − 0.701·45-s + 0.859·47-s − 0.706·49-s + 0.718·51-s − 1.42·53-s + 2.57·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 2.04T + 3T^{2} \) |
| 5 | \( 1 + 4.02T + 5T^{2} \) |
| 7 | \( 1 - 1.43T + 7T^{2} \) |
| 11 | \( 1 + 4.74T + 11T^{2} \) |
| 13 | \( 1 - 2.52T + 13T^{2} \) |
| 17 | \( 1 - 2.51T + 17T^{2} \) |
| 19 | \( 1 - 3.36T + 19T^{2} \) |
| 23 | \( 1 - 2.10T + 23T^{2} \) |
| 29 | \( 1 - 3.73T + 29T^{2} \) |
| 31 | \( 1 + 9.98T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 11.3T + 41T^{2} \) |
| 43 | \( 1 + 2.86T + 43T^{2} \) |
| 47 | \( 1 - 5.89T + 47T^{2} \) |
| 53 | \( 1 + 10.3T + 53T^{2} \) |
| 59 | \( 1 - 3.29T + 59T^{2} \) |
| 61 | \( 1 + 13.7T + 61T^{2} \) |
| 67 | \( 1 - 0.398T + 67T^{2} \) |
| 71 | \( 1 + 13.2T + 71T^{2} \) |
| 73 | \( 1 + 7.28T + 73T^{2} \) |
| 79 | \( 1 + 5.83T + 79T^{2} \) |
| 83 | \( 1 + 11.1T + 83T^{2} \) |
| 89 | \( 1 + 5.60T + 89T^{2} \) |
| 97 | \( 1 + 3.96T + 97T^{2} \) |
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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
−7.966084349395161639346163591531, −7.72245968910640125022511246042, −7.07283425584736991257420968724, −5.71257860580836179392756630722, −4.87127022993263855926308167806, −4.10355617584874380455692374932, −3.18406722419967054071530906998, −2.94295189413651405583602596488, −1.46994697637734016786734831278, 0,
1.46994697637734016786734831278, 2.94295189413651405583602596488, 3.18406722419967054071530906998, 4.10355617584874380455692374932, 4.87127022993263855926308167806, 5.71257860580836179392756630722, 7.07283425584736991257420968724, 7.72245968910640125022511246042, 7.966084349395161639346163591531
