| L(s) = 1 | − 2-s + 1.12·3-s + 4-s + 3.70·5-s − 1.12·6-s − 8-s − 1.73·9-s − 3.70·10-s + 4.57·11-s + 1.12·12-s + 5.55·13-s + 4.17·15-s + 16-s − 1.43·17-s + 1.73·18-s + 3.55·19-s + 3.70·20-s − 4.57·22-s + 5.55·23-s − 1.12·24-s + 8.72·25-s − 5.55·26-s − 5.32·27-s + 6.40·29-s − 4.17·30-s + 0.172·31-s − 32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.650·3-s + 0.5·4-s + 1.65·5-s − 0.459·6-s − 0.353·8-s − 0.577·9-s − 1.17·10-s + 1.38·11-s + 0.325·12-s + 1.53·13-s + 1.07·15-s + 0.250·16-s − 0.347·17-s + 0.408·18-s + 0.814·19-s + 0.828·20-s − 0.976·22-s + 1.15·23-s − 0.229·24-s + 1.74·25-s − 1.08·26-s − 1.02·27-s + 1.18·29-s − 0.761·30-s + 0.0309·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.868997172\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.868997172\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 7 | \( 1 \) |
| 41 | \( 1 + T \) |
| good | 3 | \( 1 - 1.12T + 3T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 11 | \( 1 - 4.57T + 11T^{2} \) |
| 13 | \( 1 - 5.55T + 13T^{2} \) |
| 17 | \( 1 + 1.43T + 17T^{2} \) |
| 19 | \( 1 - 3.55T + 19T^{2} \) |
| 23 | \( 1 - 5.55T + 23T^{2} \) |
| 29 | \( 1 - 6.40T + 29T^{2} \) |
| 31 | \( 1 - 0.172T + 31T^{2} \) |
| 37 | \( 1 - 0.252T + 37T^{2} \) |
| 43 | \( 1 + 6.42T + 43T^{2} \) |
| 47 | \( 1 + 8.11T + 47T^{2} \) |
| 53 | \( 1 + 8.41T + 53T^{2} \) |
| 59 | \( 1 + 9.87T + 59T^{2} \) |
| 61 | \( 1 + 14.3T + 61T^{2} \) |
| 67 | \( 1 - 7.13T + 67T^{2} \) |
| 71 | \( 1 - 5.81T + 71T^{2} \) |
| 73 | \( 1 + 1.38T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 14.9T + 89T^{2} \) |
| 97 | \( 1 + 9.58T + 97T^{2} \) |
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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
−8.691255901457325643853563103877, −8.013515486443773867834192680971, −6.77693669200284377193087114000, −6.38240861085711131792542229527, −5.77039032199070057510986749065, −4.78305747720690970591650361224, −3.42402980181061592095421217470, −2.88869070423227516131653320100, −1.71195429302917401116177430183, −1.20988649380053312958506922556,
1.20988649380053312958506922556, 1.71195429302917401116177430183, 2.88869070423227516131653320100, 3.42402980181061592095421217470, 4.78305747720690970591650361224, 5.77039032199070057510986749065, 6.38240861085711131792542229527, 6.77693669200284377193087114000, 8.013515486443773867834192680971, 8.691255901457325643853563103877
