| L(s) = 1 | − 2·2-s + 4·4-s + 17.3·5-s − 26.0·7-s − 8·8-s − 34.6·10-s + 4.22·11-s + 64.0·13-s + 52.1·14-s + 16·16-s + 48.5·17-s + 19·19-s + 69.2·20-s − 8.45·22-s − 92.0·23-s + 174.·25-s − 128.·26-s − 104.·28-s + 88.2·29-s − 81.9·31-s − 32·32-s − 97.0·34-s − 451.·35-s − 23.6·37-s − 38·38-s − 138.·40-s − 17.7·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.54·5-s − 1.40·7-s − 0.353·8-s − 1.09·10-s + 0.115·11-s + 1.36·13-s + 0.996·14-s + 0.250·16-s + 0.692·17-s + 0.229·19-s + 0.774·20-s − 0.0819·22-s − 0.834·23-s + 1.39·25-s − 0.966·26-s − 0.704·28-s + 0.564·29-s − 0.474·31-s − 0.176·32-s − 0.489·34-s − 2.18·35-s − 0.104·37-s − 0.162·38-s − 0.547·40-s − 0.0674·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.643968248\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.643968248\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 19T \) |
| good | 5 | \( 1 - 17.3T + 125T^{2} \) |
| 7 | \( 1 + 26.0T + 343T^{2} \) |
| 11 | \( 1 - 4.22T + 1.33e3T^{2} \) |
| 13 | \( 1 - 64.0T + 2.19e3T^{2} \) |
| 17 | \( 1 - 48.5T + 4.91e3T^{2} \) |
| 23 | \( 1 + 92.0T + 1.21e4T^{2} \) |
| 29 | \( 1 - 88.2T + 2.43e4T^{2} \) |
| 31 | \( 1 + 81.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 23.6T + 5.06e4T^{2} \) |
| 41 | \( 1 + 17.7T + 6.89e4T^{2} \) |
| 43 | \( 1 - 368.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 497.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 536.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 36.6T + 2.05e5T^{2} \) |
| 61 | \( 1 - 630.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 282.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 595.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 597.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 427.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 493.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 921.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.08e3T + 9.12e5T^{2} \) |
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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
−10.65201819912436956140692357176, −10.06017360147365565022137271214, −9.354299466658243507023825448626, −8.620793712541040335464464802991, −7.16115153892809889298238966100, −6.12662002360359603957431684000, −5.75413205458427732528581100133, −3.63377345169242378504607717245, −2.39467410093462703388209044592, −1.00292859078687388485417763120,
1.00292859078687388485417763120, 2.39467410093462703388209044592, 3.63377345169242378504607717245, 5.75413205458427732528581100133, 6.12662002360359603957431684000, 7.16115153892809889298238966100, 8.620793712541040335464464802991, 9.354299466658243507023825448626, 10.06017360147365565022137271214, 10.65201819912436956140692357176
