L(s) = 1 | + 0.0493·2-s + 3-s − 1.99·4-s + 5-s + 0.0493·6-s + 2.05·7-s − 0.197·8-s + 9-s + 0.0493·10-s − 2.48·11-s − 1.99·12-s − 4.54·13-s + 0.101·14-s + 15-s + 3.98·16-s + 0.213·17-s + 0.0493·18-s + 3.24·19-s − 1.99·20-s + 2.05·21-s − 0.122·22-s − 4.81·23-s − 0.197·24-s + 25-s − 0.224·26-s + 27-s − 4.09·28-s + ⋯ |
L(s) = 1 | + 0.0348·2-s + 0.577·3-s − 0.998·4-s + 0.447·5-s + 0.0201·6-s + 0.774·7-s − 0.0696·8-s + 0.333·9-s + 0.0155·10-s − 0.749·11-s − 0.576·12-s − 1.26·13-s + 0.0270·14-s + 0.258·15-s + 0.996·16-s + 0.0518·17-s + 0.0116·18-s + 0.745·19-s − 0.446·20-s + 0.447·21-s − 0.0261·22-s − 1.00·23-s − 0.0402·24-s + 0.200·25-s − 0.0439·26-s + 0.192·27-s − 0.773·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 401 | \( 1 + T \) |
good | 2 | \( 1 - 0.0493T + 2T^{2} \) |
| 7 | \( 1 - 2.05T + 7T^{2} \) |
| 11 | \( 1 + 2.48T + 11T^{2} \) |
| 13 | \( 1 + 4.54T + 13T^{2} \) |
| 17 | \( 1 - 0.213T + 17T^{2} \) |
| 19 | \( 1 - 3.24T + 19T^{2} \) |
| 23 | \( 1 + 4.81T + 23T^{2} \) |
| 29 | \( 1 + 0.162T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 2.70T + 37T^{2} \) |
| 41 | \( 1 - 1.86T + 41T^{2} \) |
| 43 | \( 1 + 6.03T + 43T^{2} \) |
| 47 | \( 1 - 9.97T + 47T^{2} \) |
| 53 | \( 1 + 5.57T + 53T^{2} \) |
| 59 | \( 1 + 7.70T + 59T^{2} \) |
| 61 | \( 1 + 7.15T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 + 0.128T + 73T^{2} \) |
| 79 | \( 1 - 0.411T + 79T^{2} \) |
| 83 | \( 1 - 11.4T + 83T^{2} \) |
| 89 | \( 1 + 14.8T + 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−7.76038094114860521585306628850, −7.40566640127697323913460391454, −6.16341646470352548571313829726, −5.32862746432551370009878974750, −4.87447649169374283581143451402, −4.17100386451458002858016457314, −3.18478512094667590316262315013, −2.35742340477810366551440948811, −1.41371796112523207460556335628, 0,
1.41371796112523207460556335628, 2.35742340477810366551440948811, 3.18478512094667590316262315013, 4.17100386451458002858016457314, 4.87447649169374283581143451402, 5.32862746432551370009878974750, 6.16341646470352548571313829726, 7.40566640127697323913460391454, 7.76038094114860521585306628850