| L(s) = 1 | + 2·2-s − 3·3-s + 4·4-s − 20.8·5-s − 6·6-s + 8·8-s + 9·9-s − 41.6·10-s + 15.1·11-s − 12·12-s − 2.16·13-s + 62.5·15-s + 16·16-s + 119.·17-s + 18·18-s + 33.5·19-s − 83.3·20-s + 30.3·22-s + 0.651·23-s − 24·24-s + 309.·25-s − 4.32·26-s − 27·27-s − 163.·29-s + 125.·30-s + 223.·31-s + 32·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 1.86·5-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.31·10-s + 0.415·11-s − 0.288·12-s − 0.0461·13-s + 1.07·15-s + 0.250·16-s + 1.70·17-s + 0.235·18-s + 0.404·19-s − 0.931·20-s + 0.293·22-s + 0.00590·23-s − 0.204·24-s + 2.47·25-s − 0.0326·26-s − 0.192·27-s − 1.04·29-s + 0.760·30-s + 1.29·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.700359911\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.700359911\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 + 3T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 20.8T + 125T^{2} \) |
| 11 | \( 1 - 15.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 2.16T + 2.19e3T^{2} \) |
| 17 | \( 1 - 119.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 33.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 0.651T + 1.21e4T^{2} \) |
| 29 | \( 1 + 163.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 223.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 168.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 323.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 221.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 508.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 176.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 454.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 38.6T + 2.26e5T^{2} \) |
| 67 | \( 1 - 141.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 602.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.10e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 116.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 568.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 383.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 334.T + 9.12e5T^{2} \) |
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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
−11.54258345595655558889648991570, −10.87464253335143202706370729904, −9.592182095334407446812386647386, −8.006582193505075356085977511652, −7.51267993705453352825755030030, −6.34258580245333526702382353627, −5.09068099252254613152878113813, −4.09168588254448070600844980953, −3.22026884602611775519971206382, −0.868904609939446211252465652118,
0.868904609939446211252465652118, 3.22026884602611775519971206382, 4.09168588254448070600844980953, 5.09068099252254613152878113813, 6.34258580245333526702382353627, 7.51267993705453352825755030030, 8.006582193505075356085977511652, 9.592182095334407446812386647386, 10.87464253335143202706370729904, 11.54258345595655558889648991570
