| L(s) = 1 | − 1.44·2-s − 5.92·4-s + 6.60·5-s + 20.0·8-s − 9.52·10-s − 29.1·11-s − 16.7·13-s + 18.4·16-s + 47.1·17-s − 3.43·19-s − 39.0·20-s + 42.0·22-s + 29.5·23-s − 81.3·25-s + 24.2·26-s + 223.·29-s − 120.·31-s − 187.·32-s − 67.9·34-s + 34.9·37-s + 4.94·38-s + 132.·40-s − 192.·41-s − 5.61·43-s + 172.·44-s − 42.5·46-s + 359.·47-s + ⋯ |
| L(s) = 1 | − 0.509·2-s − 0.740·4-s + 0.590·5-s + 0.887·8-s − 0.301·10-s − 0.799·11-s − 0.358·13-s + 0.287·16-s + 0.672·17-s − 0.0414·19-s − 0.437·20-s + 0.407·22-s + 0.267·23-s − 0.651·25-s + 0.182·26-s + 1.43·29-s − 0.699·31-s − 1.03·32-s − 0.342·34-s + 0.155·37-s + 0.0211·38-s + 0.523·40-s − 0.735·41-s − 0.0198·43-s + 0.591·44-s − 0.136·46-s + 1.11·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.44T + 8T^{2} \) |
| 5 | \( 1 - 6.60T + 125T^{2} \) |
| 11 | \( 1 + 29.1T + 1.33e3T^{2} \) |
| 13 | \( 1 + 16.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 47.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 3.43T + 6.85e3T^{2} \) |
| 23 | \( 1 - 29.5T + 1.21e4T^{2} \) |
| 29 | \( 1 - 223.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 120.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 34.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + 192.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 5.61T + 7.95e4T^{2} \) |
| 47 | \( 1 - 359.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 33.2T + 1.48e5T^{2} \) |
| 59 | \( 1 + 742.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 658.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 941.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 871.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 732.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.34e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 588.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.24e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 691.T + 9.12e5T^{2} \) |
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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
−8.991049538446179567816209841001, −8.045628715864446918129525596428, −7.53352136462074041137734691045, −6.35181922857790541529983545127, −5.36653417760125764188263931036, −4.76968205501847693766139252687, −3.59520190683690185664445209989, −2.39999774326099236956391765512, −1.19561187508319816207266681447, 0,
1.19561187508319816207266681447, 2.39999774326099236956391765512, 3.59520190683690185664445209989, 4.76968205501847693766139252687, 5.36653417760125764188263931036, 6.35181922857790541529983545127, 7.53352136462074041137734691045, 8.045628715864446918129525596428, 8.991049538446179567816209841001
