| L(s) = 1 | + 1.60·2-s − 3-s + 0.583·4-s + 5-s − 1.60·6-s + 0.620·7-s − 2.27·8-s + 9-s + 1.60·10-s + 4.20·11-s − 0.583·12-s + 1.55·13-s + 0.997·14-s − 15-s − 4.82·16-s − 1.20·17-s + 1.60·18-s + 8.14·19-s + 0.583·20-s − 0.620·21-s + 6.75·22-s − 2.59·23-s + 2.27·24-s + 25-s + 2.49·26-s − 27-s + 0.362·28-s + ⋯ |
| L(s) = 1 | + 1.13·2-s − 0.577·3-s + 0.291·4-s + 0.447·5-s − 0.656·6-s + 0.234·7-s − 0.805·8-s + 0.333·9-s + 0.508·10-s + 1.26·11-s − 0.168·12-s + 0.430·13-s + 0.266·14-s − 0.258·15-s − 1.20·16-s − 0.293·17-s + 0.378·18-s + 1.86·19-s + 0.130·20-s − 0.135·21-s + 1.44·22-s − 0.541·23-s + 0.464·24-s + 0.200·25-s + 0.489·26-s − 0.192·27-s + 0.0684·28-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
| \(L(1)\) |
\(\approx\) |
\(3.378049798\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.378049798\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.60T + 2T^{2} \) |
| 7 | \( 1 - 0.620T + 7T^{2} \) |
| 11 | \( 1 - 4.20T + 11T^{2} \) |
| 13 | \( 1 - 1.55T + 13T^{2} \) |
| 17 | \( 1 + 1.20T + 17T^{2} \) |
| 19 | \( 1 - 8.14T + 19T^{2} \) |
| 23 | \( 1 + 2.59T + 23T^{2} \) |
| 29 | \( 1 + 2.80T + 29T^{2} \) |
| 31 | \( 1 - 1.41T + 31T^{2} \) |
| 37 | \( 1 + 2.48T + 37T^{2} \) |
| 41 | \( 1 - 4.20T + 41T^{2} \) |
| 43 | \( 1 - 3.10T + 43T^{2} \) |
| 47 | \( 1 - 1.78T + 47T^{2} \) |
| 53 | \( 1 - 7.70T + 53T^{2} \) |
| 59 | \( 1 + 1.69T + 59T^{2} \) |
| 61 | \( 1 + 11.8T + 61T^{2} \) |
| 67 | \( 1 - 12.7T + 67T^{2} \) |
| 71 | \( 1 + 4.30T + 71T^{2} \) |
| 73 | \( 1 + 14.5T + 73T^{2} \) |
| 79 | \( 1 - 8.16T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 15.9T + 89T^{2} \) |
| 97 | \( 1 - 0.268T + 97T^{2} \) |
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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
−7.917062996361722926495040704827, −7.07722007937916863158456831505, −6.35041509930135590302513455918, −5.84644055510379808661443184824, −5.22647605702341413218833823797, −4.50091260437137838287601838378, −3.79834046001955986883999488656, −3.10970840787059476692870148584, −1.90001066635985274247431445431, −0.876042031966386716023319474283,
0.876042031966386716023319474283, 1.90001066635985274247431445431, 3.10970840787059476692870148584, 3.79834046001955986883999488656, 4.50091260437137838287601838378, 5.22647605702341413218833823797, 5.84644055510379808661443184824, 6.35041509930135590302513455918, 7.07722007937916863158456831505, 7.917062996361722926495040704827
