| L(s) = 1 | + 2.48·3-s + 4.02·5-s − 7-s + 3.18·9-s − 1.23·11-s − 6.47·13-s + 10.0·15-s + 17-s + 6.97·19-s − 2.48·21-s + 7.07·23-s + 11.1·25-s + 0.462·27-s − 6.31·29-s + 4.88·31-s − 3.07·33-s − 4.02·35-s − 2.63·37-s − 16.0·39-s − 10.1·41-s − 2.69·43-s + 12.8·45-s − 3.67·47-s + 49-s + 2.48·51-s − 1.39·53-s − 4.97·55-s + ⋯ |
| L(s) = 1 | + 1.43·3-s + 1.79·5-s − 0.377·7-s + 1.06·9-s − 0.372·11-s − 1.79·13-s + 2.58·15-s + 0.242·17-s + 1.60·19-s − 0.542·21-s + 1.47·23-s + 2.23·25-s + 0.0890·27-s − 1.17·29-s + 0.877·31-s − 0.535·33-s − 0.680·35-s − 0.433·37-s − 2.57·39-s − 1.59·41-s − 0.411·43-s + 1.91·45-s − 0.536·47-s + 0.142·49-s + 0.348·51-s − 0.190·53-s − 0.670·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.151762324\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.151762324\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| good | 3 | \( 1 - 2.48T + 3T^{2} \) |
| 5 | \( 1 - 4.02T + 5T^{2} \) |
| 11 | \( 1 + 1.23T + 11T^{2} \) |
| 13 | \( 1 + 6.47T + 13T^{2} \) |
| 19 | \( 1 - 6.97T + 19T^{2} \) |
| 23 | \( 1 - 7.07T + 23T^{2} \) |
| 29 | \( 1 + 6.31T + 29T^{2} \) |
| 31 | \( 1 - 4.88T + 31T^{2} \) |
| 37 | \( 1 + 2.63T + 37T^{2} \) |
| 41 | \( 1 + 10.1T + 41T^{2} \) |
| 43 | \( 1 + 2.69T + 43T^{2} \) |
| 47 | \( 1 + 3.67T + 47T^{2} \) |
| 53 | \( 1 + 1.39T + 53T^{2} \) |
| 59 | \( 1 - 5.07T + 59T^{2} \) |
| 61 | \( 1 - 3.93T + 61T^{2} \) |
| 67 | \( 1 + 5.63T + 67T^{2} \) |
| 71 | \( 1 + 6.84T + 71T^{2} \) |
| 73 | \( 1 - 0.246T + 73T^{2} \) |
| 79 | \( 1 + 11.6T + 79T^{2} \) |
| 83 | \( 1 + 6.65T + 83T^{2} \) |
| 89 | \( 1 - 2.27T + 89T^{2} \) |
| 97 | \( 1 - 10.7T + 97T^{2} \) |
| 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
−9.755367061872343840194135694247, −9.417130090382842666880462234609, −8.596890656848833942347992688502, −7.45573674703440886486363731111, −6.88318493055012144071278069086, −5.53519093636602594069865187673, −4.94755820588135288222236203543, −3.16994094336016536604075659221, −2.66957999933762524855464894900, −1.65067954048848718462117741045,
1.65067954048848718462117741045, 2.66957999933762524855464894900, 3.16994094336016536604075659221, 4.94755820588135288222236203543, 5.53519093636602594069865187673, 6.88318493055012144071278069086, 7.45573674703440886486363731111, 8.596890656848833942347992688502, 9.417130090382842666880462234609, 9.755367061872343840194135694247
