| L(s) = 1 | + 2.08·2-s + 2.33·4-s − 0.145·5-s + 4.41·7-s + 0.697·8-s − 0.302·10-s − 5.48·11-s + 7.15·13-s + 9.19·14-s − 3.21·16-s + 2.53·17-s + 6.32·19-s − 0.339·20-s − 11.4·22-s + 6.14·23-s − 4.97·25-s + 14.8·26-s + 10.3·28-s + 3.45·29-s + 2.61·31-s − 8.09·32-s + 5.28·34-s − 0.642·35-s − 5.11·37-s + 13.1·38-s − 0.101·40-s − 8.14·41-s + ⋯ |
| L(s) = 1 | + 1.47·2-s + 1.16·4-s − 0.0650·5-s + 1.66·7-s + 0.246·8-s − 0.0957·10-s − 1.65·11-s + 1.98·13-s + 2.45·14-s − 0.804·16-s + 0.615·17-s + 1.45·19-s − 0.0759·20-s − 2.43·22-s + 1.28·23-s − 0.995·25-s + 2.92·26-s + 1.94·28-s + 0.641·29-s + 0.469·31-s − 1.43·32-s + 0.906·34-s − 0.108·35-s − 0.840·37-s + 2.13·38-s − 0.0160·40-s − 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.795721947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.795721947\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 - 2.08T + 2T^{2} \) |
| 5 | \( 1 + 0.145T + 5T^{2} \) |
| 7 | \( 1 - 4.41T + 7T^{2} \) |
| 11 | \( 1 + 5.48T + 11T^{2} \) |
| 13 | \( 1 - 7.15T + 13T^{2} \) |
| 17 | \( 1 - 2.53T + 17T^{2} \) |
| 19 | \( 1 - 6.32T + 19T^{2} \) |
| 23 | \( 1 - 6.14T + 23T^{2} \) |
| 29 | \( 1 - 3.45T + 29T^{2} \) |
| 31 | \( 1 - 2.61T + 31T^{2} \) |
| 37 | \( 1 + 5.11T + 37T^{2} \) |
| 41 | \( 1 + 8.14T + 41T^{2} \) |
| 43 | \( 1 - 2.27T + 43T^{2} \) |
| 47 | \( 1 + 10.7T + 47T^{2} \) |
| 53 | \( 1 + 2.29T + 53T^{2} \) |
| 59 | \( 1 + 6.11T + 59T^{2} \) |
| 61 | \( 1 - 0.756T + 61T^{2} \) |
| 67 | \( 1 - 5.05T + 67T^{2} \) |
| 71 | \( 1 - 7.11T + 71T^{2} \) |
| 73 | \( 1 - 14.1T + 73T^{2} \) |
| 79 | \( 1 - 5.71T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 - 0.443T + 89T^{2} \) |
| 97 | \( 1 + 9.59T + 97T^{2} \) |
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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
−8.154463118228936737566255766141, −7.31764676876453860070251679394, −6.40453955965672959621793741677, −5.55030814855789240462674253290, −5.13302595590985297126930319556, −4.72056085296688468714396099523, −3.57837933195761708965820910859, −3.17162489292283642061389326200, −2.05518972701051381091067963822, −1.09761517192219446340648195673,
1.09761517192219446340648195673, 2.05518972701051381091067963822, 3.17162489292283642061389326200, 3.57837933195761708965820910859, 4.72056085296688468714396099523, 5.13302595590985297126930319556, 5.55030814855789240462674253290, 6.40453955965672959621793741677, 7.31764676876453860070251679394, 8.154463118228936737566255766141
