L(s) = 1 | − 2-s + 1.61·3-s + 4-s − 1.61·6-s − 1.85·7-s − 8-s − 0.381·9-s − 5.61·11-s + 1.61·12-s + 2.61·13-s + 1.85·14-s + 16-s + 0.854·17-s + 0.381·18-s − 0.145·19-s − 3·21-s + 5.61·22-s + 23-s − 1.61·24-s − 2.61·26-s − 5.47·27-s − 1.85·28-s − 9.70·29-s − 2.14·31-s − 32-s − 9.09·33-s − 0.854·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.934·3-s + 0.5·4-s − 0.660·6-s − 0.700·7-s − 0.353·8-s − 0.127·9-s − 1.69·11-s + 0.467·12-s + 0.726·13-s + 0.495·14-s + 0.250·16-s + 0.207·17-s + 0.0900·18-s − 0.0334·19-s − 0.654·21-s + 1.19·22-s + 0.208·23-s − 0.330·24-s − 0.513·26-s − 1.05·27-s − 0.350·28-s − 1.80·29-s − 0.385·31-s − 0.176·32-s − 1.58·33-s − 0.146·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 - 1.61T + 3T^{2} \) |
| 7 | \( 1 + 1.85T + 7T^{2} \) |
| 11 | \( 1 + 5.61T + 11T^{2} \) |
| 13 | \( 1 - 2.61T + 13T^{2} \) |
| 17 | \( 1 - 0.854T + 17T^{2} \) |
| 19 | \( 1 + 0.145T + 19T^{2} \) |
| 29 | \( 1 + 9.70T + 29T^{2} \) |
| 31 | \( 1 + 2.14T + 31T^{2} \) |
| 37 | \( 1 + 9.70T + 37T^{2} \) |
| 41 | \( 1 + 5.61T + 41T^{2} \) |
| 43 | \( 1 - 11.2T + 43T^{2} \) |
| 47 | \( 1 - 1.70T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 2.85T + 61T^{2} \) |
| 67 | \( 1 - 5.23T + 67T^{2} \) |
| 71 | \( 1 - 0.381T + 71T^{2} \) |
| 73 | \( 1 + 16.4T + 73T^{2} \) |
| 79 | \( 1 + 7.70T + 79T^{2} \) |
| 83 | \( 1 - 7.70T + 83T^{2} \) |
| 89 | \( 1 - 3.70T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−9.187385740378998336598095626375, −8.701486361567116497147555109629, −7.80802235881064409621797475313, −7.31069633849413177281321705342, −6.06621505915817515721636923887, −5.29226553569133621347239092469, −3.65256573034507040884604499670, −2.95478052463480280464988825522, −1.95836919282468872674729701641, 0,
1.95836919282468872674729701641, 2.95478052463480280464988825522, 3.65256573034507040884604499670, 5.29226553569133621347239092469, 6.06621505915817515721636923887, 7.31069633849413177281321705342, 7.80802235881064409621797475313, 8.701486361567116497147555109629, 9.187385740378998336598095626375