| L(s) = 1 | − 2.60·2-s + 4.78·4-s − 1.68·5-s + 2.80·7-s − 7.25·8-s + 4.37·10-s + 1.84·11-s − 0.773·13-s − 7.31·14-s + 9.33·16-s + 0.882·17-s − 3.94·19-s − 8.04·20-s − 4.81·22-s + 23-s − 2.17·25-s + 2.01·26-s + 13.4·28-s − 29-s + 5.16·31-s − 9.79·32-s − 2.29·34-s − 4.71·35-s + 3.00·37-s + 10.2·38-s + 12.1·40-s − 1.65·41-s + ⋯ |
| L(s) = 1 | − 1.84·2-s + 2.39·4-s − 0.751·5-s + 1.06·7-s − 2.56·8-s + 1.38·10-s + 0.557·11-s − 0.214·13-s − 1.95·14-s + 2.33·16-s + 0.214·17-s − 0.904·19-s − 1.79·20-s − 1.02·22-s + 0.208·23-s − 0.435·25-s + 0.395·26-s + 2.53·28-s − 0.185·29-s + 0.928·31-s − 1.73·32-s − 0.394·34-s − 0.797·35-s + 0.493·37-s + 1.66·38-s + 1.92·40-s − 0.257·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6003 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| good | 2 | \( 1 + 2.60T + 2T^{2} \) |
| 5 | \( 1 + 1.68T + 5T^{2} \) |
| 7 | \( 1 - 2.80T + 7T^{2} \) |
| 11 | \( 1 - 1.84T + 11T^{2} \) |
| 13 | \( 1 + 0.773T + 13T^{2} \) |
| 17 | \( 1 - 0.882T + 17T^{2} \) |
| 19 | \( 1 + 3.94T + 19T^{2} \) |
| 31 | \( 1 - 5.16T + 31T^{2} \) |
| 37 | \( 1 - 3.00T + 37T^{2} \) |
| 41 | \( 1 + 1.65T + 41T^{2} \) |
| 43 | \( 1 + 3.35T + 43T^{2} \) |
| 47 | \( 1 - 1.98T + 47T^{2} \) |
| 53 | \( 1 + 1.44T + 53T^{2} \) |
| 59 | \( 1 + 2.83T + 59T^{2} \) |
| 61 | \( 1 - 4.60T + 61T^{2} \) |
| 67 | \( 1 + 13.2T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 12.2T + 73T^{2} \) |
| 79 | \( 1 + 2.29T + 79T^{2} \) |
| 83 | \( 1 + 6.46T + 83T^{2} \) |
| 89 | \( 1 + 16.1T + 89T^{2} \) |
| 97 | \( 1 - 11.0T + 97T^{2} \) |
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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
−7.87248810661761536155994889543, −7.39578464589048924778250381518, −6.65094357464082581959990907770, −5.92928765953403474575018503707, −4.78686013976796308452492255039, −3.97984257848315384078346355354, −2.85434940652243265484989406391, −1.91529676856108422278251664711, −1.15214417600015922551248989246, 0,
1.15214417600015922551248989246, 1.91529676856108422278251664711, 2.85434940652243265484989406391, 3.97984257848315384078346355354, 4.78686013976796308452492255039, 5.92928765953403474575018503707, 6.65094357464082581959990907770, 7.39578464589048924778250381518, 7.87248810661761536155994889543
