L(s) = 1 | + 1.47·2-s + 3-s + 0.187·4-s + 5-s + 1.47·6-s − 0.812·7-s − 2.68·8-s + 9-s + 1.47·10-s + 4.15·11-s + 0.187·12-s + 6.68·13-s − 1.20·14-s + 15-s − 4.33·16-s + 7.87·17-s + 1.47·18-s − 5.72·19-s + 0.187·20-s − 0.812·21-s + 6.15·22-s − 2.68·24-s + 25-s + 9.88·26-s + 27-s − 0.152·28-s + 3.34·29-s + ⋯ |
L(s) = 1 | + 1.04·2-s + 0.577·3-s + 0.0935·4-s + 0.447·5-s + 0.603·6-s − 0.307·7-s − 0.947·8-s + 0.333·9-s + 0.467·10-s + 1.25·11-s + 0.0540·12-s + 1.85·13-s − 0.321·14-s + 0.258·15-s − 1.08·16-s + 1.90·17-s + 0.348·18-s − 1.31·19-s + 0.0418·20-s − 0.177·21-s + 1.31·22-s − 0.547·24-s + 0.200·25-s + 1.93·26-s + 0.192·27-s − 0.0287·28-s + 0.621·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7935 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.142594766\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.142594766\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 \) |
good | 2 | \( 1 - 1.47T + 2T^{2} \) |
| 7 | \( 1 + 0.812T + 7T^{2} \) |
| 11 | \( 1 - 4.15T + 11T^{2} \) |
| 13 | \( 1 - 6.68T + 13T^{2} \) |
| 17 | \( 1 - 7.87T + 17T^{2} \) |
| 19 | \( 1 + 5.72T + 19T^{2} \) |
| 29 | \( 1 - 3.34T + 29T^{2} \) |
| 31 | \( 1 + 2.81T + 31T^{2} \) |
| 37 | \( 1 - 6.67T + 37T^{2} \) |
| 41 | \( 1 + 7.87T + 41T^{2} \) |
| 43 | \( 1 + 6.84T + 43T^{2} \) |
| 47 | \( 1 - 5.78T + 47T^{2} \) |
| 53 | \( 1 + 4.97T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 - 8.68T + 61T^{2} \) |
| 67 | \( 1 + 0.576T + 67T^{2} \) |
| 71 | \( 1 - 0.885T + 71T^{2} \) |
| 73 | \( 1 - 0.770T + 73T^{2} \) |
| 79 | \( 1 - 14.5T + 79T^{2} \) |
| 83 | \( 1 - 10.7T + 83T^{2} \) |
| 89 | \( 1 - 3.33T + 89T^{2} \) |
| 97 | \( 1 - 6.05T + 97T^{2} \) |
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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
−8.034121124065062364424193194710, −6.75298408331891800166130812449, −6.31557315327762121557985607191, −5.84953377272432925628449226321, −4.95710773547502898635161652113, −4.10667553413459311070644251918, −3.53260552424961917606167751017, −3.14210637111958450566986263420, −1.88261464354146609879582094540, −1.00559613931281874013951819507,
1.00559613931281874013951819507, 1.88261464354146609879582094540, 3.14210637111958450566986263420, 3.53260552424961917606167751017, 4.10667553413459311070644251918, 4.95710773547502898635161652113, 5.84953377272432925628449226321, 6.31557315327762121557985607191, 6.75298408331891800166130812449, 8.034121124065062364424193194710