| L(s) = 1 | + 1.64·2-s − 5.30·4-s + 14.9·5-s + 8.99·7-s − 21.8·8-s + 24.5·10-s + 38.0·11-s + 48.4·13-s + 14.7·14-s + 6.54·16-s + 67.1·17-s − 150.·19-s − 79.3·20-s + 62.5·22-s + 178.·23-s + 98.7·25-s + 79.6·26-s − 47.6·28-s + 207.·29-s + 3.02·31-s + 185.·32-s + 110.·34-s + 134.·35-s − 202.·37-s − 246.·38-s − 326.·40-s − 137.·41-s + ⋯ |
| L(s) = 1 | + 0.580·2-s − 0.662·4-s + 1.33·5-s + 0.485·7-s − 0.965·8-s + 0.776·10-s + 1.04·11-s + 1.03·13-s + 0.281·14-s + 0.102·16-s + 0.958·17-s − 1.81·19-s − 0.886·20-s + 0.606·22-s + 1.62·23-s + 0.790·25-s + 0.600·26-s − 0.321·28-s + 1.32·29-s + 0.0175·31-s + 1.02·32-s + 0.556·34-s + 0.649·35-s − 0.900·37-s − 1.05·38-s − 1.29·40-s − 0.524·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2151 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(4.134324772\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.134324772\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 239 | \( 1 - 239T \) |
| good | 2 | \( 1 - 1.64T + 8T^{2} \) |
| 5 | \( 1 - 14.9T + 125T^{2} \) |
| 7 | \( 1 - 8.99T + 343T^{2} \) |
| 11 | \( 1 - 38.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 48.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 67.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 150.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 178.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 207.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 3.02T + 2.97e4T^{2} \) |
| 37 | \( 1 + 202.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 137.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 3.56T + 7.95e4T^{2} \) |
| 47 | \( 1 + 29.9T + 1.03e5T^{2} \) |
| 53 | \( 1 - 199.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 6.04T + 2.05e5T^{2} \) |
| 61 | \( 1 + 75.2T + 2.26e5T^{2} \) |
| 67 | \( 1 - 1.01e3T + 3.00e5T^{2} \) |
| 71 | \( 1 + 326.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 533.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 416.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 22.6T + 5.71e5T^{2} \) |
| 89 | \( 1 - 950.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.23e3T + 9.12e5T^{2} \) |
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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
−8.737830726266520996263774933787, −8.304179204386368157523186795776, −6.74168663105822296647131333277, −6.29147436841249131270602261772, −5.48487434018142727910984420439, −4.79052446178373634186389921557, −3.91403631972622110501372421117, −3.00050739165003562619099618798, −1.75554663745781780190167112929, −0.918576106422944849409351070260,
0.918576106422944849409351070260, 1.75554663745781780190167112929, 3.00050739165003562619099618798, 3.91403631972622110501372421117, 4.79052446178373634186389921557, 5.48487434018142727910984420439, 6.29147436841249131270602261772, 6.74168663105822296647131333277, 8.304179204386368157523186795776, 8.737830726266520996263774933787
