| L(s) = 1 | − 5.88·2-s + 2.58·4-s − 25·5-s + 111.·7-s + 172.·8-s + 147.·10-s + 268.·11-s + 169·13-s − 658.·14-s − 1.10e3·16-s + 866.·17-s + 1.92e3·19-s − 64.7·20-s − 1.57e3·22-s + 3.45e3·23-s + 625·25-s − 993.·26-s + 289.·28-s + 4.51e3·29-s + 3.09e3·31-s + 935.·32-s − 5.09e3·34-s − 2.79e3·35-s + 5.41e3·37-s − 1.13e4·38-s − 4.32e3·40-s + 1.70e4·41-s + ⋯ |
| L(s) = 1 | − 1.03·2-s + 0.0809·4-s − 0.447·5-s + 0.863·7-s + 0.955·8-s + 0.464·10-s + 0.668·11-s + 0.277·13-s − 0.897·14-s − 1.07·16-s + 0.727·17-s + 1.22·19-s − 0.0361·20-s − 0.695·22-s + 1.36·23-s + 0.200·25-s − 0.288·26-s + 0.0698·28-s + 0.995·29-s + 0.578·31-s + 0.161·32-s − 0.755·34-s − 0.386·35-s + 0.649·37-s − 1.27·38-s − 0.427·40-s + 1.58·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.483770597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.483770597\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + 25T \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 5.88T + 32T^{2} \) |
| 7 | \( 1 - 111.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 268.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 866.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.92e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.45e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.51e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 3.09e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.70e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.48e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.30e3T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.67e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 5.48e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.14e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 499.T + 1.35e9T^{2} \) |
| 71 | \( 1 - 2.09e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.41e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.74e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.37e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 6.16e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 2.54e4T + 8.58e9T^{2} \) |
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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
−9.714314989795697995245231795697, −9.068095099259430986270889717645, −8.132457853253654512599397868407, −7.65205715605135507405378517473, −6.62064709915268842346301324912, −5.17331170355229422871662369788, −4.38818590481494022743780114023, −3.10217745760828372436353898985, −1.41873260906253871151925968857, −0.815601707011801470873009025113,
0.815601707011801470873009025113, 1.41873260906253871151925968857, 3.10217745760828372436353898985, 4.38818590481494022743780114023, 5.17331170355229422871662369788, 6.62064709915268842346301324912, 7.65205715605135507405378517473, 8.132457853253654512599397868407, 9.068095099259430986270889717645, 9.714314989795697995245231795697
