| L(s) = 1 | − 0.311·2-s − 3-s − 1.90·4-s + 0.311·6-s − 0.688·7-s + 1.21·8-s + 9-s + 2·11-s + 1.90·12-s − 3.73·13-s + 0.214·14-s + 3.42·16-s − 2.28·17-s − 0.311·18-s + 0.688·21-s − 0.622·22-s − 2.90·23-s − 1.21·24-s + 1.16·26-s − 27-s + 1.31·28-s − 4.02·29-s + 31-s − 3.49·32-s − 2·33-s + 0.709·34-s − 1.90·36-s + ⋯ |
| L(s) = 1 | − 0.219·2-s − 0.577·3-s − 0.951·4-s + 0.127·6-s − 0.260·7-s + 0.429·8-s + 0.333·9-s + 0.603·11-s + 0.549·12-s − 1.03·13-s + 0.0572·14-s + 0.857·16-s − 0.553·17-s − 0.0733·18-s + 0.150·21-s − 0.132·22-s − 0.605·23-s − 0.247·24-s + 0.228·26-s − 0.192·27-s + 0.247·28-s − 0.746·29-s + 0.179·31-s − 0.617·32-s − 0.348·33-s + 0.121·34-s − 0.317·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2325 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6977687290\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6977687290\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 31 | \( 1 - T \) |
| good | 2 | \( 1 + 0.311T + 2T^{2} \) |
| 7 | \( 1 + 0.688T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 3.73T + 13T^{2} \) |
| 17 | \( 1 + 2.28T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 + 4.02T + 29T^{2} \) |
| 37 | \( 1 - 8.79T + 37T^{2} \) |
| 41 | \( 1 - 2.75T + 41T^{2} \) |
| 43 | \( 1 + 1.05T + 43T^{2} \) |
| 47 | \( 1 + 4.70T + 47T^{2} \) |
| 53 | \( 1 + 2.14T + 53T^{2} \) |
| 59 | \( 1 + 6.44T + 59T^{2} \) |
| 61 | \( 1 + 5.05T + 61T^{2} \) |
| 67 | \( 1 - 5.44T + 67T^{2} \) |
| 71 | \( 1 - 8.96T + 71T^{2} \) |
| 73 | \( 1 - 8.92T + 73T^{2} \) |
| 79 | \( 1 + 8.38T + 79T^{2} \) |
| 83 | \( 1 - 10.5T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
| 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
−9.337392318742662434836557272870, −8.188323658244873343753629104435, −7.56889361259129142412019915715, −6.60186635150314609670440299228, −5.87392535238770335842723932578, −4.89293122486444706365892845695, −4.36310457369763917021805410422, −3.38494487690111702847105673042, −1.96844776596237429350183663319, −0.57118620944822108919533441319,
0.57118620944822108919533441319, 1.96844776596237429350183663319, 3.38494487690111702847105673042, 4.36310457369763917021805410422, 4.89293122486444706365892845695, 5.87392535238770335842723932578, 6.60186635150314609670440299228, 7.56889361259129142412019915715, 8.188323658244873343753629104435, 9.337392318742662434836557272870
