L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 7-s − 8-s + 9-s + 10-s − 1.05·11-s − 12-s − 6.49·13-s + 14-s + 15-s + 16-s + 3.55·17-s − 18-s + 7.43·19-s − 20-s + 21-s + 1.05·22-s + 23-s + 24-s + 25-s + 6.49·26-s − 27-s − 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 − 0.377·7-s − 0.353·8-s + 0.333·9-s + 0.316·10-s − 0.319·11-s − 0.288·12-s − 1.80·13-s + 0.267·14-s + 0.258·15-s + 0.250·16-s + 0.862·17-s − 0.235·18-s + 1.70·19-s − 0.223·20-s + 0.218·21-s + 0.225·22-s + 0.208·23-s + 0.204·24-s + 0.200·25-s + 1.27·26-s − 0.192·27-s − 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5696023182\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5696023182\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 11 | \( 1 + 1.05T + 11T^{2} \) |
| 13 | \( 1 + 6.49T + 13T^{2} \) |
| 17 | \( 1 - 3.55T + 17T^{2} \) |
| 19 | \( 1 - 7.43T + 19T^{2} \) |
| 29 | \( 1 + 3.05T + 29T^{2} \) |
| 31 | \( 1 + 1.55T + 31T^{2} \) |
| 37 | \( 1 + 6T + 37T^{2} \) |
| 41 | \( 1 - 0.117T + 41T^{2} \) |
| 43 | \( 1 + 2.94T + 43T^{2} \) |
| 47 | \( 1 + 7.67T + 47T^{2} \) |
| 53 | \( 1 + 6T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 0.117T + 61T^{2} \) |
| 67 | \( 1 - 5.05T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 + 9.11T + 73T^{2} \) |
| 79 | \( 1 - 13.2T + 79T^{2} \) |
| 83 | \( 1 - 3.43T + 83T^{2} \) |
| 89 | \( 1 - 4.61T + 89T^{2} \) |
| 97 | \( 1 - 17.6T + 97T^{2} \) |
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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
−8.086312119877532865359671486916, −7.42102418924534849201062728461, −7.19196672631098165633892372584, −6.19175907716140730196047859189, −5.25967855307535695907041034955, −4.88337306998514636256510479667, −3.53003298467526060061190956391, −2.89047301485959874892664372594, −1.69131285135751201756916777830, −0.47152191771368251456268293615,
0.47152191771368251456268293615, 1.69131285135751201756916777830, 2.89047301485959874892664372594, 3.53003298467526060061190956391, 4.88337306998514636256510479667, 5.25967855307535695907041034955, 6.19175907716140730196047859189, 7.19196672631098165633892372584, 7.42102418924534849201062728461, 8.086312119877532865359671486916