| L(s) = 1 | + 4.87·2-s − 4.14·3-s + 15.7·4-s − 20.2·6-s + 37.6·8-s − 9.78·9-s + 36.9·11-s − 65.2·12-s − 61.3·13-s + 57.7·16-s − 44.8·17-s − 47.6·18-s + 139.·19-s + 180.·22-s − 217.·23-s − 156.·24-s − 298.·26-s + 152.·27-s − 33.8·29-s − 124.·31-s − 20.2·32-s − 153.·33-s − 218.·34-s − 153.·36-s − 237.·37-s + 680.·38-s + 254.·39-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 0.798·3-s + 1.96·4-s − 1.37·6-s + 1.66·8-s − 0.362·9-s + 1.01·11-s − 1.57·12-s − 1.30·13-s + 0.902·16-s − 0.639·17-s − 0.624·18-s + 1.68·19-s + 1.74·22-s − 1.97·23-s − 1.33·24-s − 2.25·26-s + 1.08·27-s − 0.216·29-s − 0.720·31-s − 0.111·32-s − 0.809·33-s − 1.10·34-s − 0.712·36-s − 1.05·37-s + 2.90·38-s + 1.04·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - 4.87T + 8T^{2} \) |
| 3 | \( 1 + 4.14T + 27T^{2} \) |
| 11 | \( 1 - 36.9T + 1.33e3T^{2} \) |
| 13 | \( 1 + 61.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 44.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 139.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 217.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 33.8T + 2.43e4T^{2} \) |
| 31 | \( 1 + 124.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 237.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 343.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 16.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 346.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 135.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 490.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 477.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 45.2T + 3.57e5T^{2} \) |
| 73 | \( 1 - 100.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 880.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 619.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 231.T + 9.12e5T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.067176368169110199107479587714, −7.66588133764104473851040105360, −6.90577747017518638876188848603, −6.10266817647913224128306225781, −5.47377215237084821810903600770, −4.74744492993242245056402390935, −3.88066246383256787242973696272, −2.90755977607247918896845824918, −1.76817491752824106549789722053, 0,
1.76817491752824106549789722053, 2.90755977607247918896845824918, 3.88066246383256787242973696272, 4.74744492993242245056402390935, 5.47377215237084821810903600770, 6.10266817647913224128306225781, 6.90577747017518638876188848603, 7.66588133764104473851040105360, 9.067176368169110199107479587714
