| L(s) = 1 | + 2-s + 1.18·3-s + 4-s − 5-s + 1.18·6-s + 8-s − 1.58·9-s − 10-s + 3.88·11-s + 1.18·12-s + 4.93·13-s − 1.18·15-s + 16-s − 17-s − 1.58·18-s + 0.599·19-s − 20-s + 3.88·22-s + 8.10·23-s + 1.18·24-s + 25-s + 4.93·26-s − 5.45·27-s − 2.81·29-s − 1.18·30-s − 5.25·31-s + 32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.686·3-s + 0.5·4-s − 0.447·5-s + 0.485·6-s + 0.353·8-s − 0.528·9-s − 0.316·10-s + 1.17·11-s + 0.343·12-s + 1.36·13-s − 0.307·15-s + 0.250·16-s − 0.242·17-s − 0.373·18-s + 0.137·19-s − 0.223·20-s + 0.828·22-s + 1.68·23-s + 0.242·24-s + 0.200·25-s + 0.968·26-s − 1.04·27-s − 0.523·29-s − 0.217·30-s − 0.943·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.474070853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.474070853\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.18T + 3T^{2} \) |
| 11 | \( 1 - 3.88T + 11T^{2} \) |
| 13 | \( 1 - 4.93T + 13T^{2} \) |
| 19 | \( 1 - 0.599T + 19T^{2} \) |
| 23 | \( 1 - 8.10T + 23T^{2} \) |
| 29 | \( 1 + 2.81T + 29T^{2} \) |
| 31 | \( 1 + 5.25T + 31T^{2} \) |
| 37 | \( 1 + 8.25T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 - 10.7T + 43T^{2} \) |
| 47 | \( 1 + 6.09T + 47T^{2} \) |
| 53 | \( 1 + 3.62T + 53T^{2} \) |
| 59 | \( 1 - 5.61T + 59T^{2} \) |
| 61 | \( 1 - 2.50T + 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 - 16.1T + 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 - 6.37T + 83T^{2} \) |
| 89 | \( 1 + 3.22T + 89T^{2} \) |
| 97 | \( 1 - 0.283T + 97T^{2} \) |
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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
−7.70059164287993717949696406287, −7.11646606809870772407567913250, −6.35716706107357214070071777536, −5.76541690232065308725796487848, −4.94430636461181314072958824254, −3.97299680386506481058026017942, −3.60171269736225213955125470537, −2.93540565321096319327324482009, −1.91548611331154399579508487040, −0.940146080448687074065154656350,
0.940146080448687074065154656350, 1.91548611331154399579508487040, 2.93540565321096319327324482009, 3.60171269736225213955125470537, 3.97299680386506481058026017942, 4.94430636461181314072958824254, 5.76541690232065308725796487848, 6.35716706107357214070071777536, 7.11646606809870772407567913250, 7.70059164287993717949696406287
