L(s) = 1 | − 2-s + 4-s + 3.69·5-s − 8-s − 3.69·10-s + 1.47·11-s − 2.69·13-s + 16-s − 6.57·17-s − 0.888·19-s + 3.69·20-s − 1.47·22-s − 6.28·23-s + 8.68·25-s + 2.69·26-s + 2.51·29-s − 6.81·31-s − 32-s + 6.57·34-s + 2.77·37-s + 0.888·38-s − 3.69·40-s + 4.11·41-s − 0.0123·43-s + 1.47·44-s + 6.28·46-s − 6.98·47-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.65·5-s − 0.353·8-s − 1.16·10-s + 0.445·11-s − 0.748·13-s + 0.250·16-s − 1.59·17-s − 0.203·19-s + 0.827·20-s − 0.314·22-s − 1.31·23-s + 1.73·25-s + 0.529·26-s + 0.466·29-s − 1.22·31-s − 0.176·32-s + 1.12·34-s + 0.456·37-s + 0.144·38-s − 0.584·40-s + 0.642·41-s − 0.00188·43-s + 0.222·44-s + 0.927·46-s − 1.01·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7938 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 - 3.69T + 5T^{2} \) |
| 11 | \( 1 - 1.47T + 11T^{2} \) |
| 13 | \( 1 + 2.69T + 13T^{2} \) |
| 17 | \( 1 + 6.57T + 17T^{2} \) |
| 19 | \( 1 + 0.888T + 19T^{2} \) |
| 23 | \( 1 + 6.28T + 23T^{2} \) |
| 29 | \( 1 - 2.51T + 29T^{2} \) |
| 31 | \( 1 + 6.81T + 31T^{2} \) |
| 37 | \( 1 - 2.77T + 37T^{2} \) |
| 41 | \( 1 - 4.11T + 41T^{2} \) |
| 43 | \( 1 + 0.0123T + 43T^{2} \) |
| 47 | \( 1 + 6.98T + 47T^{2} \) |
| 53 | \( 1 + 3.21T + 53T^{2} \) |
| 59 | \( 1 - 6.90T + 59T^{2} \) |
| 61 | \( 1 - 5.73T + 61T^{2} \) |
| 67 | \( 1 + 9.46T + 67T^{2} \) |
| 71 | \( 1 - 5.46T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 + 4.47T + 83T^{2} \) |
| 89 | \( 1 - 8.87T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−7.47455914085131843675533146072, −6.66414478368763677712766328637, −6.30579572397566991721128840178, −5.58761506743550376660138262727, −4.80375760202406052407185214004, −3.91077886064879235016995870676, −2.62085860071902581664451044346, −2.15546072295428746472679898339, −1.43153216856913565649301385660, 0,
1.43153216856913565649301385660, 2.15546072295428746472679898339, 2.62085860071902581664451044346, 3.91077886064879235016995870676, 4.80375760202406052407185214004, 5.58761506743550376660138262727, 6.30579572397566991721128840178, 6.66414478368763677712766328637, 7.47455914085131843675533146072