| L(s) = 1 | + 2.65·2-s + 3-s + 5.05·4-s − 5-s + 2.65·6-s + 3.87·7-s + 8.12·8-s + 9-s − 2.65·10-s − 5.60·11-s + 5.05·12-s + 4.59·13-s + 10.2·14-s − 15-s + 11.4·16-s + 7.71·17-s + 2.65·18-s − 4.50·19-s − 5.05·20-s + 3.87·21-s − 14.8·22-s + 3.82·23-s + 8.12·24-s + 25-s + 12.2·26-s + 27-s + 19.5·28-s + ⋯ |
| L(s) = 1 | + 1.87·2-s + 0.577·3-s + 2.52·4-s − 0.447·5-s + 1.08·6-s + 1.46·7-s + 2.87·8-s + 0.333·9-s − 0.840·10-s − 1.69·11-s + 1.46·12-s + 1.27·13-s + 2.75·14-s − 0.258·15-s + 2.86·16-s + 1.87·17-s + 0.626·18-s − 1.03·19-s − 1.13·20-s + 0.845·21-s − 3.17·22-s + 0.797·23-s + 1.65·24-s + 0.200·25-s + 2.39·26-s + 0.192·27-s + 3.70·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6015 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.650800704\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.650800704\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 401 | \( 1 + T \) |
| good | 2 | \( 1 - 2.65T + 2T^{2} \) |
| 7 | \( 1 - 3.87T + 7T^{2} \) |
| 11 | \( 1 + 5.60T + 11T^{2} \) |
| 13 | \( 1 - 4.59T + 13T^{2} \) |
| 17 | \( 1 - 7.71T + 17T^{2} \) |
| 19 | \( 1 + 4.50T + 19T^{2} \) |
| 23 | \( 1 - 3.82T + 23T^{2} \) |
| 29 | \( 1 + 8.25T + 29T^{2} \) |
| 31 | \( 1 + 6.47T + 31T^{2} \) |
| 37 | \( 1 - 6.69T + 37T^{2} \) |
| 41 | \( 1 + 2.44T + 41T^{2} \) |
| 43 | \( 1 - 4.90T + 43T^{2} \) |
| 47 | \( 1 + 7.61T + 47T^{2} \) |
| 53 | \( 1 + 12.3T + 53T^{2} \) |
| 59 | \( 1 - 0.741T + 59T^{2} \) |
| 61 | \( 1 + 10.2T + 61T^{2} \) |
| 67 | \( 1 - 7.65T + 67T^{2} \) |
| 71 | \( 1 - 3.92T + 71T^{2} \) |
| 73 | \( 1 - 8.80T + 73T^{2} \) |
| 79 | \( 1 - 1.11T + 79T^{2} \) |
| 83 | \( 1 - 10.3T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−7.85107832274186511116176001750, −7.52110248801396281961752769029, −6.44678984705703395353812195421, −5.51651563794222281046173401516, −5.21767907986677191757752140916, −4.44191075992654300997160717038, −3.66394661374918193009518899359, −3.11930080012551318053579096563, −2.16645007860700053369406126617, −1.39915187996550436250206979724,
1.39915187996550436250206979724, 2.16645007860700053369406126617, 3.11930080012551318053579096563, 3.66394661374918193009518899359, 4.44191075992654300997160717038, 5.21767907986677191757752140916, 5.51651563794222281046173401516, 6.44678984705703395353812195421, 7.52110248801396281961752769029, 7.85107832274186511116176001750
