| L(s) = 1 | − 2-s + 2.41·3-s + 4-s − 5-s − 2.41·6-s + 5.18·7-s − 8-s + 2.82·9-s + 10-s − 5.27·11-s + 2.41·12-s − 5.86·13-s − 5.18·14-s − 2.41·15-s + 16-s − 4.21·17-s − 2.82·18-s − 1.98·19-s − 20-s + 12.5·21-s + 5.27·22-s − 2.41·24-s + 25-s + 5.86·26-s − 0.414·27-s + 5.18·28-s − 0.889·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.39·3-s + 0.5·4-s − 0.447·5-s − 0.985·6-s + 1.95·7-s − 0.353·8-s + 0.942·9-s + 0.316·10-s − 1.59·11-s + 0.696·12-s − 1.62·13-s − 1.38·14-s − 0.623·15-s + 0.250·16-s − 1.02·17-s − 0.666·18-s − 0.454·19-s − 0.223·20-s + 2.72·21-s + 1.12·22-s − 0.492·24-s + 0.200·25-s + 1.14·26-s − 0.0797·27-s + 0.979·28-s − 0.165·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5290 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 - 2.41T + 3T^{2} \) |
| 7 | \( 1 - 5.18T + 7T^{2} \) |
| 11 | \( 1 + 5.27T + 11T^{2} \) |
| 13 | \( 1 + 5.86T + 13T^{2} \) |
| 17 | \( 1 + 4.21T + 17T^{2} \) |
| 19 | \( 1 + 1.98T + 19T^{2} \) |
| 29 | \( 1 + 0.889T + 29T^{2} \) |
| 31 | \( 1 - 6.68T + 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 + 3.39T + 41T^{2} \) |
| 43 | \( 1 - 3.19T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 + 3.24T + 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 - 4.09T + 61T^{2} \) |
| 67 | \( 1 + 10.6T + 67T^{2} \) |
| 71 | \( 1 + 6.90T + 71T^{2} \) |
| 73 | \( 1 - 4.44T + 73T^{2} \) |
| 79 | \( 1 + 17.4T + 79T^{2} \) |
| 83 | \( 1 - 3.74T + 83T^{2} \) |
| 89 | \( 1 + 10.7T + 89T^{2} \) |
| 97 | \( 1 - 1.11T + 97T^{2} \) |
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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
−7.77774675120034201634134009647, −7.67937801808955558683940713660, −6.91880924161180320157591648933, −5.45419765306206997224846017087, −4.78625657030757738741973702121, −4.17995131586427718592027796652, −2.75606427170556199835218931385, −2.43793357958384430190599156545, −1.64204627053946441333242904458, 0,
1.64204627053946441333242904458, 2.43793357958384430190599156545, 2.75606427170556199835218931385, 4.17995131586427718592027796652, 4.78625657030757738741973702121, 5.45419765306206997224846017087, 6.91880924161180320157591648933, 7.67937801808955558683940713660, 7.77774675120034201634134009647
