| L(s) = 1 | + 2·2-s + 4·4-s + 5·5-s − 7·7-s + 8·8-s + 10·10-s + 25·11-s + 13·13-s − 14·14-s + 16·16-s − 29·17-s − 98·19-s + 20·20-s + 50·22-s − 133·23-s + 25·25-s + 26·26-s − 28·28-s − 217·29-s − 237·31-s + 32·32-s − 58·34-s − 35·35-s + 186·37-s − 196·38-s + 40·40-s + 24·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.377·7-s + 0.353·8-s + 0.316·10-s + 0.685·11-s + 0.277·13-s − 0.267·14-s + 1/4·16-s − 0.413·17-s − 1.18·19-s + 0.223·20-s + 0.484·22-s − 1.20·23-s + 1/5·25-s + 0.196·26-s − 0.188·28-s − 1.38·29-s − 1.37·31-s + 0.176·32-s − 0.292·34-s − 0.169·35-s + 0.826·37-s − 0.836·38-s + 0.158·40-s + 0.0914·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1890 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 7 | \( 1 + p T \) |
| good | 11 | \( 1 - 25 T + p^{3} T^{2} \) |
| 13 | \( 1 - p T + p^{3} T^{2} \) |
| 17 | \( 1 + 29 T + p^{3} T^{2} \) |
| 19 | \( 1 + 98 T + p^{3} T^{2} \) |
| 23 | \( 1 + 133 T + p^{3} T^{2} \) |
| 29 | \( 1 + 217 T + p^{3} T^{2} \) |
| 31 | \( 1 + 237 T + p^{3} T^{2} \) |
| 37 | \( 1 - 186 T + p^{3} T^{2} \) |
| 41 | \( 1 - 24 T + p^{3} T^{2} \) |
| 43 | \( 1 + 31 T + p^{3} T^{2} \) |
| 47 | \( 1 + 201 T + p^{3} T^{2} \) |
| 53 | \( 1 + 518 T + p^{3} T^{2} \) |
| 59 | \( 1 - 684 T + p^{3} T^{2} \) |
| 61 | \( 1 - 224 T + p^{3} T^{2} \) |
| 67 | \( 1 - 204 T + p^{3} T^{2} \) |
| 71 | \( 1 + 160 T + p^{3} T^{2} \) |
| 73 | \( 1 + 752 T + p^{3} T^{2} \) |
| 79 | \( 1 - 691 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1128 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1180 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1456 T + p^{3} T^{2} \) |
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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
−8.497223944068080013046714354765, −7.53609493185161286041074702157, −6.59819367457672322819446283435, −6.11457512926232452819938493748, −5.30431450460132652283688205850, −4.17579952755753712001467674034, −3.65029395874349774997965725087, −2.38476122436064365014677191146, −1.60419380223155943959221306083, 0,
1.60419380223155943959221306083, 2.38476122436064365014677191146, 3.65029395874349774997965725087, 4.17579952755753712001467674034, 5.30431450460132652283688205850, 6.11457512926232452819938493748, 6.59819367457672322819446283435, 7.53609493185161286041074702157, 8.497223944068080013046714354765
