| L(s) = 1 | + 2·2-s + 4·4-s − 5·5-s + 26·7-s + 8·8-s − 10·10-s − 54·11-s + 32·13-s + 52·14-s + 16·16-s − 78·17-s + 19·19-s − 20·20-s − 108·22-s − 12·23-s + 25·25-s + 64·26-s + 104·28-s − 204·29-s − 256·31-s + 32·32-s − 156·34-s − 130·35-s − 340·37-s + 38·38-s − 40·40-s + 156·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.447·5-s + 1.40·7-s + 0.353·8-s − 0.316·10-s − 1.48·11-s + 0.682·13-s + 0.992·14-s + 1/4·16-s − 1.11·17-s + 0.229·19-s − 0.223·20-s − 1.04·22-s − 0.108·23-s + 1/5·25-s + 0.482·26-s + 0.701·28-s − 1.30·29-s − 1.48·31-s + 0.176·32-s − 0.786·34-s − 0.627·35-s − 1.51·37-s + 0.162·38-s − 0.158·40-s + 0.594·41-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 | \( 1 - 26 T + p^{3} T^{2} \) |
| 11 | \( 1 + 54 T + p^{3} T^{2} \) |
| 13 | \( 1 - 32 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 23 | \( 1 + 12 T + p^{3} T^{2} \) |
| 29 | \( 1 + 204 T + p^{3} T^{2} \) |
| 31 | \( 1 + 256 T + p^{3} T^{2} \) |
| 37 | \( 1 + 340 T + p^{3} T^{2} \) |
| 41 | \( 1 - 156 T + p^{3} T^{2} \) |
| 43 | \( 1 - 326 T + p^{3} T^{2} \) |
| 47 | \( 1 - 132 T + p^{3} T^{2} \) |
| 53 | \( 1 + 90 T + p^{3} T^{2} \) |
| 59 | \( 1 - 360 T + p^{3} T^{2} \) |
| 61 | \( 1 + 838 T + p^{3} T^{2} \) |
| 67 | \( 1 + 16 T + p^{3} T^{2} \) |
| 71 | \( 1 + 888 T + p^{3} T^{2} \) |
| 73 | \( 1 - 854 T + p^{3} T^{2} \) |
| 79 | \( 1 + 640 T + p^{3} T^{2} \) |
| 83 | \( 1 - 84 T + p^{3} T^{2} \) |
| 89 | \( 1 + 828 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1424 T + p^{3} T^{2} \) |
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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
−8.417412870727895084359850647044, −7.65185703591879723442205571727, −7.17390549093879643862413300248, −5.84340698434921625833981346454, −5.23901767358919425045109873688, −4.47301322755718109356658018574, −3.63870562274982983013951066445, −2.44582576877928197150354223500, −1.58761506463118262732198000337, 0,
1.58761506463118262732198000337, 2.44582576877928197150354223500, 3.63870562274982983013951066445, 4.47301322755718109356658018574, 5.23901767358919425045109873688, 5.84340698434921625833981346454, 7.17390549093879643862413300248, 7.65185703591879723442205571727, 8.417412870727895084359850647044
