L(s) = 1 | − 2.63·2-s − 2.36·3-s + 4.92·4-s + 6.21·6-s + 2.69·7-s − 7.70·8-s + 2.57·9-s + 2.27·11-s − 11.6·12-s + 6.58·13-s − 7.09·14-s + 10.4·16-s + 4.62·17-s − 6.78·18-s + 5.32·19-s − 6.36·21-s − 5.98·22-s − 1.94·23-s + 18.1·24-s − 17.3·26-s + 1.00·27-s + 13.2·28-s + 5.77·29-s + 1.90·31-s − 12.0·32-s − 5.36·33-s − 12.1·34-s + ⋯ |
L(s) = 1 | − 1.86·2-s − 1.36·3-s + 2.46·4-s + 2.53·6-s + 1.01·7-s − 2.72·8-s + 0.858·9-s + 0.685·11-s − 3.35·12-s + 1.82·13-s − 1.89·14-s + 2.60·16-s + 1.12·17-s − 1.59·18-s + 1.22·19-s − 1.38·21-s − 1.27·22-s − 0.405·23-s + 3.71·24-s − 3.40·26-s + 0.192·27-s + 2.51·28-s + 1.07·29-s + 0.341·31-s − 2.12·32-s − 0.934·33-s − 2.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6099197412\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6099197412\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 47 | \( 1 - T \) |
good | 2 | \( 1 + 2.63T + 2T^{2} \) |
| 3 | \( 1 + 2.36T + 3T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 - 2.27T + 11T^{2} \) |
| 13 | \( 1 - 6.58T + 13T^{2} \) |
| 17 | \( 1 - 4.62T + 17T^{2} \) |
| 19 | \( 1 - 5.32T + 19T^{2} \) |
| 23 | \( 1 + 1.94T + 23T^{2} \) |
| 29 | \( 1 - 5.77T + 29T^{2} \) |
| 31 | \( 1 - 1.90T + 31T^{2} \) |
| 37 | \( 1 - 8.51T + 37T^{2} \) |
| 41 | \( 1 + 9.83T + 41T^{2} \) |
| 43 | \( 1 + 10.3T + 43T^{2} \) |
| 53 | \( 1 - 3.08T + 53T^{2} \) |
| 59 | \( 1 + 0.754T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 + 7.00T + 67T^{2} \) |
| 71 | \( 1 + 1.77T + 71T^{2} \) |
| 73 | \( 1 + 0.970T + 73T^{2} \) |
| 79 | \( 1 - 13.8T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 - 10.3T + 89T^{2} \) |
| 97 | \( 1 - 15.9T + 97T^{2} \) |
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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
−9.955852549234876444250089720548, −8.886156656955687723330603761356, −8.231792876705237077119849865364, −7.52249535960872026119214583265, −6.43806734097566338687542462687, −6.01804667225692479692987907528, −4.92415407719283800962083187680, −3.33433756780463339521613336357, −1.48083907290381705998919930051, −0.967937319768426894256973076274,
0.967937319768426894256973076274, 1.48083907290381705998919930051, 3.33433756780463339521613336357, 4.92415407719283800962083187680, 6.01804667225692479692987907528, 6.43806734097566338687542462687, 7.52249535960872026119214583265, 8.231792876705237077119849865364, 8.886156656955687723330603761356, 9.955852549234876444250089720548