L(s) = 1 | + 2-s + 4-s + 3.23·5-s − 7-s + 8-s + 3.23·10-s − 6.47·13-s − 14-s + 16-s + 2.76·17-s + 4.47·19-s + 3.23·20-s + 23-s + 5.47·25-s − 6.47·26-s − 28-s + 2·29-s + 3.23·31-s + 32-s + 2.76·34-s − 3.23·35-s + 6·37-s + 4.47·38-s + 3.23·40-s + 10·41-s + 2.47·43-s + 46-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s + 1.44·5-s − 0.377·7-s + 0.353·8-s + 1.02·10-s − 1.79·13-s − 0.267·14-s + 0.250·16-s + 0.670·17-s + 1.02·19-s + 0.723·20-s + 0.208·23-s + 1.09·25-s − 1.26·26-s − 0.188·28-s + 0.371·29-s + 0.581·31-s + 0.176·32-s + 0.474·34-s − 0.546·35-s + 0.986·37-s + 0.725·38-s + 0.511·40-s + 1.56·41-s + 0.376·43-s + 0.147·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.711629389\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.711629389\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 5 | \( 1 - 3.23T + 5T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 17 | \( 1 - 2.76T + 17T^{2} \) |
| 19 | \( 1 - 4.47T + 19T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 3.23T + 31T^{2} \) |
| 37 | \( 1 - 6T + 37T^{2} \) |
| 41 | \( 1 - 10T + 41T^{2} \) |
| 43 | \( 1 - 2.47T + 43T^{2} \) |
| 47 | \( 1 - 11.2T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 6.76T + 59T^{2} \) |
| 61 | \( 1 + 1.70T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 6.47T + 71T^{2} \) |
| 73 | \( 1 - 13.4T + 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 + 10.9T + 83T^{2} \) |
| 89 | \( 1 + 6.18T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−9.011232051352668053479860286714, −7.69866297603610462479234142356, −7.20095061210415025914491298722, −6.26682397318652874660922930294, −5.62739557677466252773303188339, −5.07368544097290215752507173254, −4.14663407077729232926573451558, −2.77455627253228564364406023046, −2.49038445678904597617129358442, −1.13016948224013605500194388943,
1.13016948224013605500194388943, 2.49038445678904597617129358442, 2.77455627253228564364406023046, 4.14663407077729232926573451558, 5.07368544097290215752507173254, 5.62739557677466252773303188339, 6.26682397318652874660922930294, 7.20095061210415025914491298722, 7.69866297603610462479234142356, 9.011232051352668053479860286714