L(s) = 1 | − 1.37·2-s + 1.78·3-s + 0.883·4-s − 2.45·6-s − 1.67·7-s + 0.159·8-s + 2.19·9-s + 1.57·12-s + 2.29·14-s − 1.10·16-s − 3.01·18-s + 1.94·19-s − 2.98·21-s + 0.792·23-s + 0.285·24-s + 25-s + 2.13·27-s − 1.47·28-s + 1.53·31-s + 1.35·32-s + 1.93·36-s + 0.347·37-s − 2.67·38-s − 1.87·41-s + 4.09·42-s − 0.573·43-s − 1.08·46-s + ⋯ |
L(s) = 1 | − 1.37·2-s + 1.78·3-s + 0.883·4-s − 2.45·6-s − 1.67·7-s + 0.159·8-s + 2.19·9-s + 1.57·12-s + 2.29·14-s − 1.10·16-s − 3.01·18-s + 1.94·19-s − 2.98·21-s + 0.792·23-s + 0.285·24-s + 25-s + 2.13·27-s − 1.47·28-s + 1.53·31-s + 1.35·32-s + 1.93·36-s + 0.347·37-s − 2.67·38-s − 1.87·41-s + 4.09·42-s − 0.573·43-s − 1.08·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 983 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8440564275\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8440564275\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 983 | \( 1 - T \) |
good | 2 | \( 1 + 1.37T + T^{2} \) |
| 3 | \( 1 - 1.78T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 7 | \( 1 + 1.67T + T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.94T + T^{2} \) |
| 23 | \( 1 - 0.792T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 1.53T + T^{2} \) |
| 37 | \( 1 - 0.347T + T^{2} \) |
| 41 | \( 1 + 1.87T + T^{2} \) |
| 43 | \( 1 + 0.573T + T^{2} \) |
| 47 | \( 1 + 1.98T + T^{2} \) |
| 53 | \( 1 + 0.573T + T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.87T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - T^{2} \) |
| 79 | \( 1 + T + T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + 0.116T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.868370229949521779758990481178, −9.324423042822212614711527920955, −8.726978323157110778308482279410, −7.961830351700021071727134410722, −7.16120820787735991125831752546, −6.57017234488807257448519548231, −4.74137747761227584140947625415, −3.23779736752656169076878360319, −2.95298206511947889534990129428, −1.37418132463718640727369348763,
1.37418132463718640727369348763, 2.95298206511947889534990129428, 3.23779736752656169076878360319, 4.74137747761227584140947625415, 6.57017234488807257448519548231, 7.16120820787735991125831752546, 7.961830351700021071727134410722, 8.726978323157110778308482279410, 9.324423042822212614711527920955, 9.868370229949521779758990481178