| L(s) = 1 | + 2-s + 0.414·3-s + 4-s + 5-s + 0.414·6-s + 8-s − 2.82·9-s + 10-s + 0.585·11-s + 0.414·12-s − 6.24·13-s + 0.414·15-s + 16-s − 2.17·17-s − 2.82·18-s + 2.82·19-s + 20-s + 0.585·22-s − 4·23-s + 0.414·24-s − 4·25-s − 6.24·26-s − 2.41·27-s + 3.82·29-s + 0.414·30-s − 8.41·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.239·3-s + 0.5·4-s + 0.447·5-s + 0.169·6-s + 0.353·8-s − 0.942·9-s + 0.316·10-s + 0.176·11-s + 0.119·12-s − 1.73·13-s + 0.106·15-s + 0.250·16-s − 0.526·17-s − 0.666·18-s + 0.648·19-s + 0.223·20-s + 0.124·22-s − 0.834·23-s + 0.0845·24-s − 0.800·25-s − 1.22·26-s − 0.464·27-s + 0.710·29-s + 0.0756·30-s − 1.51·31-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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 \) |
| 41 | \( 1 - T \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 5 | \( 1 - T + 5T^{2} \) |
| 11 | \( 1 - 0.585T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 + 2.17T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 - 3.82T + 29T^{2} \) |
| 31 | \( 1 + 8.41T + 31T^{2} \) |
| 37 | \( 1 + 1.41T + 37T^{2} \) |
| 43 | \( 1 + 6.07T + 43T^{2} \) |
| 47 | \( 1 + 8.58T + 47T^{2} \) |
| 53 | \( 1 - 8.65T + 53T^{2} \) |
| 59 | \( 1 - 4.82T + 59T^{2} \) |
| 61 | \( 1 + 2.65T + 61T^{2} \) |
| 67 | \( 1 - 3.17T + 67T^{2} \) |
| 71 | \( 1 - 1.24T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 3.24T + 79T^{2} \) |
| 83 | \( 1 + 1.07T + 83T^{2} \) |
| 89 | \( 1 + 7.82T + 89T^{2} \) |
| 97 | \( 1 + 12.1T + 97T^{2} \) |
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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
−7.981840531003402806705916867385, −7.26313743166387675291827169467, −6.55718246167725626755629749097, −5.63512599019283696962963362508, −5.21825054062601776129222257047, −4.29745193231163591992311329759, −3.36084313736349966544019734523, −2.52290714788147974560207544984, −1.85623924024583279096241369081, 0,
1.85623924024583279096241369081, 2.52290714788147974560207544984, 3.36084313736349966544019734523, 4.29745193231163591992311329759, 5.21825054062601776129222257047, 5.63512599019283696962963362508, 6.55718246167725626755629749097, 7.26313743166387675291827169467, 7.981840531003402806705916867385
