L(s) = 1 | − 4.08·2-s + 8.66·4-s + 3.30·5-s + 5.57·7-s − 2.72·8-s − 13.4·10-s + 62.0·11-s + 7.80·13-s − 22.7·14-s − 58.2·16-s + 107.·17-s − 26.2·19-s + 28.6·20-s − 253.·22-s − 132.·23-s − 114.·25-s − 31.8·26-s + 48.3·28-s − 178.·29-s − 5.33·31-s + 259.·32-s − 438.·34-s + 18.4·35-s − 109.·37-s + 107.·38-s − 8.99·40-s − 144.·41-s + ⋯ |
L(s) = 1 | − 1.44·2-s + 1.08·4-s + 0.295·5-s + 0.301·7-s − 0.120·8-s − 0.426·10-s + 1.70·11-s + 0.166·13-s − 0.434·14-s − 0.909·16-s + 1.53·17-s − 0.316·19-s + 0.319·20-s − 2.45·22-s − 1.19·23-s − 0.912·25-s − 0.240·26-s + 0.326·28-s − 1.14·29-s − 0.0309·31-s + 1.43·32-s − 2.21·34-s + 0.0889·35-s − 0.488·37-s + 0.457·38-s − 0.0355·40-s − 0.550·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1503 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 167 | \( 1 + 167T \) |
good | 2 | \( 1 + 4.08T + 8T^{2} \) |
| 5 | \( 1 - 3.30T + 125T^{2} \) |
| 7 | \( 1 - 5.57T + 343T^{2} \) |
| 11 | \( 1 - 62.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 7.80T + 2.19e3T^{2} \) |
| 17 | \( 1 - 107.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 26.2T + 6.85e3T^{2} \) |
| 23 | \( 1 + 132.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 178.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 5.33T + 2.97e4T^{2} \) |
| 37 | \( 1 + 109.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 144.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 51.5T + 7.95e4T^{2} \) |
| 47 | \( 1 + 223.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 15.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 845.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 75.8T + 2.26e5T^{2} \) |
| 67 | \( 1 + 102.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 961.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.14e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.04e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 781.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 89.5T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.43e3T + 9.12e5T^{2} \) |
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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
−8.794968467654997955398868817479, −8.040445348850992573742944188617, −7.40235890585483601704760299278, −6.44992670499592339744671544950, −5.71218407933351866624936638865, −4.37594464282583488454741267820, −3.45062560215079658581120635662, −1.81881784706869627642941109302, −1.36067867799540901514112873400, 0,
1.36067867799540901514112873400, 1.81881784706869627642941109302, 3.45062560215079658581120635662, 4.37594464282583488454741267820, 5.71218407933351866624936638865, 6.44992670499592339744671544950, 7.40235890585483601704760299278, 8.040445348850992573742944188617, 8.794968467654997955398868817479