| L(s) = 1 | − 2-s + 0.0555·3-s + 4-s + 0.675·5-s − 0.0555·6-s − 0.0958·7-s − 8-s − 2.99·9-s − 0.675·10-s + 1.17·11-s + 0.0555·12-s + 5.23·13-s + 0.0958·14-s + 0.0375·15-s + 16-s + 0.451·17-s + 2.99·18-s − 0.859·19-s + 0.675·20-s − 0.00532·21-s − 1.17·22-s − 5.05·23-s − 0.0555·24-s − 4.54·25-s − 5.23·26-s − 0.333·27-s − 0.0958·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.0320·3-s + 0.5·4-s + 0.302·5-s − 0.0226·6-s − 0.0362·7-s − 0.353·8-s − 0.998·9-s − 0.213·10-s + 0.354·11-s + 0.0160·12-s + 1.45·13-s + 0.0256·14-s + 0.00969·15-s + 0.250·16-s + 0.109·17-s + 0.706·18-s − 0.197·19-s + 0.151·20-s − 0.00116·21-s − 0.250·22-s − 1.05·23-s − 0.0113·24-s − 0.908·25-s − 1.02·26-s − 0.0641·27-s − 0.0181·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 2017 | \( 1 + T \) |
| good | 3 | \( 1 - 0.0555T + 3T^{2} \) |
| 5 | \( 1 - 0.675T + 5T^{2} \) |
| 7 | \( 1 + 0.0958T + 7T^{2} \) |
| 11 | \( 1 - 1.17T + 11T^{2} \) |
| 13 | \( 1 - 5.23T + 13T^{2} \) |
| 17 | \( 1 - 0.451T + 17T^{2} \) |
| 19 | \( 1 + 0.859T + 19T^{2} \) |
| 23 | \( 1 + 5.05T + 23T^{2} \) |
| 29 | \( 1 + 1.20T + 29T^{2} \) |
| 31 | \( 1 + 8.98T + 31T^{2} \) |
| 37 | \( 1 - 4.78T + 37T^{2} \) |
| 41 | \( 1 - 3.83T + 41T^{2} \) |
| 43 | \( 1 + 1.58T + 43T^{2} \) |
| 47 | \( 1 - 2.00T + 47T^{2} \) |
| 53 | \( 1 + 6.40T + 53T^{2} \) |
| 59 | \( 1 - 6.89T + 59T^{2} \) |
| 61 | \( 1 + 6.76T + 61T^{2} \) |
| 67 | \( 1 + 4.54T + 67T^{2} \) |
| 71 | \( 1 + 3.45T + 71T^{2} \) |
| 73 | \( 1 + 0.329T + 73T^{2} \) |
| 79 | \( 1 - 11.9T + 79T^{2} \) |
| 83 | \( 1 + 11.7T + 83T^{2} \) |
| 89 | \( 1 - 16.2T + 89T^{2} \) |
| 97 | \( 1 + 0.125T + 97T^{2} \) |
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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
−8.084790572580822935370789405549, −7.63647692946501646438841748037, −6.41625162721858718357474423140, −6.07522621761211809908429481568, −5.37283670939970733947989574990, −4.02866417747042957376609356879, −3.36943356184863164517567380529, −2.26859651064479109663290337232, −1.39824317922423443039765230503, 0,
1.39824317922423443039765230503, 2.26859651064479109663290337232, 3.36943356184863164517567380529, 4.02866417747042957376609356879, 5.37283670939970733947989574990, 6.07522621761211809908429481568, 6.41625162721858718357474423140, 7.63647692946501646438841748037, 8.084790572580822935370789405549
