| L(s) = 1 | − 2.08·2-s + 2.35·4-s + 5-s + 4.08·7-s − 0.734·8-s − 2.08·10-s − 1.35·11-s + 0.648·13-s − 8.52·14-s − 3.17·16-s − 1.35·17-s + 0.648·19-s + 2.35·20-s + 2.82·22-s + 4.79·23-s + 25-s − 1.35·26-s + 9.61·28-s + 3.87·29-s − 7.69·31-s + 8.08·32-s + 2.82·34-s + 4.08·35-s + 7.52·37-s − 1.35·38-s − 0.734·40-s − 0.179·41-s + ⋯ |
| L(s) = 1 | − 1.47·2-s + 1.17·4-s + 0.447·5-s + 1.54·7-s − 0.259·8-s − 0.659·10-s − 0.407·11-s + 0.179·13-s − 2.27·14-s − 0.793·16-s − 0.327·17-s + 0.148·19-s + 0.525·20-s + 0.601·22-s + 0.998·23-s + 0.200·25-s − 0.265·26-s + 1.81·28-s + 0.719·29-s − 1.38·31-s + 1.42·32-s + 0.483·34-s + 0.690·35-s + 1.23·37-s − 0.219·38-s − 0.116·40-s − 0.0280·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8417203731\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8417203731\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 2 | \( 1 + 2.08T + 2T^{2} \) |
| 7 | \( 1 - 4.08T + 7T^{2} \) |
| 11 | \( 1 + 1.35T + 11T^{2} \) |
| 13 | \( 1 - 0.648T + 13T^{2} \) |
| 17 | \( 1 + 1.35T + 17T^{2} \) |
| 19 | \( 1 - 0.648T + 19T^{2} \) |
| 23 | \( 1 - 4.79T + 23T^{2} \) |
| 29 | \( 1 - 3.87T + 29T^{2} \) |
| 31 | \( 1 + 7.69T + 31T^{2} \) |
| 37 | \( 1 - 7.52T + 37T^{2} \) |
| 41 | \( 1 + 0.179T + 41T^{2} \) |
| 43 | \( 1 + 0.820T + 43T^{2} \) |
| 47 | \( 1 - 10.9T + 47T^{2} \) |
| 53 | \( 1 - 4.17T + 53T^{2} \) |
| 59 | \( 1 - 4.17T + 59T^{2} \) |
| 61 | \( 1 + 3.82T + 61T^{2} \) |
| 67 | \( 1 - 8.14T + 67T^{2} \) |
| 71 | \( 1 + 6.11T + 71T^{2} \) |
| 73 | \( 1 + 12.3T + 73T^{2} \) |
| 79 | \( 1 - 10.3T + 79T^{2} \) |
| 83 | \( 1 + 12.2T + 83T^{2} \) |
| 89 | \( 1 + 3T + 89T^{2} \) |
| 97 | \( 1 + 13.5T + 97T^{2} \) |
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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
−10.96848225593364782425800684115, −10.36556769895641939681826832140, −9.257898121091855577726697204488, −8.603754952090573020173189531993, −7.78788256924353841025213093669, −7.00662119091092111598756542699, −5.54882013337785530863251845106, −4.48588607705119477716971697997, −2.38904308721300866982192686263, −1.21556430091835107025965573141,
1.21556430091835107025965573141, 2.38904308721300866982192686263, 4.48588607705119477716971697997, 5.54882013337785530863251845106, 7.00662119091092111598756542699, 7.78788256924353841025213093669, 8.603754952090573020173189531993, 9.257898121091855577726697204488, 10.36556769895641939681826832140, 10.96848225593364782425800684115
