L(s) = 1 | + 2·2-s + 3·3-s + 4·4-s − 5·5-s + 6·6-s + 8.86·7-s + 8·8-s + 9·9-s − 10·10-s + 52.4·11-s + 12·12-s + 7.96·13-s + 17.7·14-s − 15·15-s + 16·16-s − 57.5·17-s + 18·18-s − 87.7·19-s − 20·20-s + 26.5·21-s + 104.·22-s + 178.·23-s + 24·24-s + 25·25-s + 15.9·26-s + 27·27-s + 35.4·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.447·5-s + 0.408·6-s + 0.478·7-s + 0.353·8-s + 0.333·9-s − 0.316·10-s + 1.43·11-s + 0.288·12-s + 0.169·13-s + 0.338·14-s − 0.258·15-s + 0.250·16-s − 0.821·17-s + 0.235·18-s − 1.05·19-s − 0.223·20-s + 0.276·21-s + 1.01·22-s + 1.61·23-s + 0.204·24-s + 0.200·25-s + 0.120·26-s + 0.192·27-s + 0.239·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 930 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(4.555870378\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.555870378\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 2T \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 + 5T \) |
| 31 | \( 1 + 31T \) |
good | 7 | \( 1 - 8.86T + 343T^{2} \) |
| 11 | \( 1 - 52.4T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.96T + 2.19e3T^{2} \) |
| 17 | \( 1 + 57.5T + 4.91e3T^{2} \) |
| 19 | \( 1 + 87.7T + 6.85e3T^{2} \) |
| 23 | \( 1 - 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 230.T + 2.43e4T^{2} \) |
| 37 | \( 1 - 17.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 162.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 258.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 150.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 239.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 496.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 212.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 118.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 878.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.17e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.21e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 323.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 32.2T + 7.04e5T^{2} \) |
| 97 | \( 1 + 820.T + 9.12e5T^{2} \) |
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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
−9.551049996602231848553371214782, −8.723063712437852073323760192644, −8.100310479630805796045945029484, −6.83151587553105123219399486075, −6.53069053733591017445360856733, −5.00189436421560810579566920507, −4.27702680502307530706643373369, −3.46290666749412070245305450514, −2.29346310119457406720244643457, −1.09367890551339101114499314039,
1.09367890551339101114499314039, 2.29346310119457406720244643457, 3.46290666749412070245305450514, 4.27702680502307530706643373369, 5.00189436421560810579566920507, 6.53069053733591017445360856733, 6.83151587553105123219399486075, 8.100310479630805796045945029484, 8.723063712437852073323760192644, 9.551049996602231848553371214782