| L(s) = 1 | + 2·2-s + 4·4-s − 5·5-s − 2·7-s + 8·8-s − 10·10-s + 16·11-s − 10·13-s − 4·14-s + 16·16-s − 36·17-s + 19·19-s − 20·20-s + 32·22-s − 124·23-s + 25·25-s − 20·26-s − 8·28-s + 174·29-s − 74·31-s + 32·32-s − 72·34-s + 10·35-s + 94·37-s + 38·38-s − 40·40-s + 240·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.107·7-s + 0.353·8-s − 0.316·10-s + 0.438·11-s − 0.213·13-s − 0.0763·14-s + 1/4·16-s − 0.513·17-s + 0.229·19-s − 0.223·20-s + 0.310·22-s − 1.12·23-s + 1/5·25-s − 0.150·26-s − 0.0539·28-s + 1.11·29-s − 0.428·31-s + 0.176·32-s − 0.363·34-s + 0.0482·35-s + 0.417·37-s + 0.162·38-s − 0.158·40-s + 0.914·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 19 | \( 1 - p T \) |
| good | 7 | \( 1 + 2 T + p^{3} T^{2} \) |
| 11 | \( 1 - 16 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 + 36 T + p^{3} T^{2} \) |
| 23 | \( 1 + 124 T + p^{3} T^{2} \) |
| 29 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 74 T + p^{3} T^{2} \) |
| 37 | \( 1 - 94 T + p^{3} T^{2} \) |
| 41 | \( 1 - 240 T + p^{3} T^{2} \) |
| 43 | \( 1 + 276 T + p^{3} T^{2} \) |
| 47 | \( 1 + 540 T + p^{3} T^{2} \) |
| 53 | \( 1 + 146 T + p^{3} T^{2} \) |
| 59 | \( 1 + 606 T + p^{3} T^{2} \) |
| 61 | \( 1 - 450 T + p^{3} T^{2} \) |
| 67 | \( 1 - 180 T + p^{3} T^{2} \) |
| 71 | \( 1 - 456 T + p^{3} T^{2} \) |
| 73 | \( 1 - 14 T + p^{3} T^{2} \) |
| 79 | \( 1 - 550 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1442 T + p^{3} T^{2} \) |
| 89 | \( 1 + 212 T + p^{3} T^{2} \) |
| 97 | \( 1 + 830 T + p^{3} T^{2} \) |
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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
−8.387315508478660411263161736343, −7.77294843169962060850748374853, −6.75953978859173441122790093968, −6.24526739324879275819379205254, −5.15364833952631973954469724529, −4.38369085238654490623807603313, −3.58849819562376178639709538743, −2.62753238992186743679936578320, −1.46268071293388553833057928997, 0,
1.46268071293388553833057928997, 2.62753238992186743679936578320, 3.58849819562376178639709538743, 4.38369085238654490623807603313, 5.15364833952631973954469724529, 6.24526739324879275819379205254, 6.75953978859173441122790093968, 7.77294843169962060850748374853, 8.387315508478660411263161736343
