L(s) = 1 | + 2-s + 0.724·3-s + 4-s − 5-s + 0.724·6-s − 1.42·7-s + 8-s − 2.47·9-s − 10-s + 1.01·11-s + 0.724·12-s + 6.23·13-s − 1.42·14-s − 0.724·15-s + 16-s − 5.95·17-s − 2.47·18-s − 0.248·19-s − 20-s − 1.03·21-s + 1.01·22-s − 9.01·23-s + 0.724·24-s + 25-s + 6.23·26-s − 3.96·27-s − 1.42·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.418·3-s + 0.5·4-s − 0.447·5-s + 0.295·6-s − 0.540·7-s + 0.353·8-s − 0.825·9-s − 0.316·10-s + 0.304·11-s + 0.209·12-s + 1.72·13-s − 0.382·14-s − 0.186·15-s + 0.250·16-s − 1.44·17-s − 0.583·18-s − 0.0569·19-s − 0.223·20-s − 0.225·21-s + 0.215·22-s − 1.88·23-s + 0.147·24-s + 0.200·25-s + 1.22·26-s − 0.763·27-s − 0.270·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 - T \) |
good | 3 | \( 1 - 0.724T + 3T^{2} \) |
| 7 | \( 1 + 1.42T + 7T^{2} \) |
| 11 | \( 1 - 1.01T + 11T^{2} \) |
| 13 | \( 1 - 6.23T + 13T^{2} \) |
| 17 | \( 1 + 5.95T + 17T^{2} \) |
| 19 | \( 1 + 0.248T + 19T^{2} \) |
| 23 | \( 1 + 9.01T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 - 2.28T + 31T^{2} \) |
| 37 | \( 1 - 10.5T + 37T^{2} \) |
| 41 | \( 1 + 9.70T + 41T^{2} \) |
| 43 | \( 1 + 7.32T + 43T^{2} \) |
| 47 | \( 1 + 0.0347T + 47T^{2} \) |
| 53 | \( 1 - 0.670T + 53T^{2} \) |
| 59 | \( 1 + 8.22T + 59T^{2} \) |
| 61 | \( 1 + 9.86T + 61T^{2} \) |
| 67 | \( 1 - 4.63T + 67T^{2} \) |
| 71 | \( 1 - 6.18T + 71T^{2} \) |
| 73 | \( 1 + 12.4T + 73T^{2} \) |
| 79 | \( 1 - 9.33T + 79T^{2} \) |
| 83 | \( 1 + 4.83T + 83T^{2} \) |
| 89 | \( 1 - 7.10T + 89T^{2} \) |
| 97 | \( 1 + 18.2T + 97T^{2} \) |
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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
−7.976579793277489200462652667009, −6.58775258647397273051589049418, −6.43037491066985576271768282083, −5.72504207222668032479515338455, −4.60214849635650163073235940964, −3.96970045051676963746111349735, −3.34305589932348898937414958742, −2.58576053308442871628406953279, −1.55185121297091713229001614161, 0,
1.55185121297091713229001614161, 2.58576053308442871628406953279, 3.34305589932348898937414958742, 3.96970045051676963746111349735, 4.60214849635650163073235940964, 5.72504207222668032479515338455, 6.43037491066985576271768282083, 6.58775258647397273051589049418, 7.976579793277489200462652667009