| L(s) = 1 | − 5.57·2-s + 23.1·4-s + 14.3·5-s − 15.9·7-s − 84.2·8-s − 80.1·10-s − 67.7·11-s − 60.3·13-s + 89.0·14-s + 285.·16-s − 112.·17-s + 14.3·19-s + 332.·20-s + 377.·22-s − 148.·23-s + 81.4·25-s + 336.·26-s − 369.·28-s − 95.1·29-s − 211.·31-s − 916.·32-s + 624.·34-s − 229.·35-s + 91.5·37-s − 80.2·38-s − 1.21e3·40-s − 330.·41-s + ⋯ |
| L(s) = 1 | − 1.97·2-s + 2.88·4-s + 1.28·5-s − 0.862·7-s − 3.72·8-s − 2.53·10-s − 1.85·11-s − 1.28·13-s + 1.70·14-s + 4.45·16-s − 1.59·17-s + 0.173·19-s + 3.71·20-s + 3.65·22-s − 1.34·23-s + 0.651·25-s + 2.53·26-s − 2.49·28-s − 0.609·29-s − 1.22·31-s − 5.06·32-s + 3.15·34-s − 1.10·35-s + 0.406·37-s − 0.342·38-s − 4.78·40-s − 1.25·41-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
| \(L(2)\) |
\(\approx\) |
\(0.003480250974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.003480250974\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 5.57T + 8T^{2} \) |
| 5 | \( 1 - 14.3T + 125T^{2} \) |
| 7 | \( 1 + 15.9T + 343T^{2} \) |
| 11 | \( 1 + 67.7T + 1.33e3T^{2} \) |
| 13 | \( 1 + 60.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 112.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 14.3T + 6.85e3T^{2} \) |
| 23 | \( 1 + 148.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 95.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 211.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 91.5T + 5.06e4T^{2} \) |
| 41 | \( 1 + 330.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 116.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 391.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 65.7T + 1.48e5T^{2} \) |
| 59 | \( 1 + 352.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 289.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 508.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 850.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 527.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 388.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 274.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.18e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.62e3T + 9.12e5T^{2} \) |
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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
−8.944322378034296914670106617319, −8.010395274096552704671255803635, −7.40405564408688640071715591227, −6.58656473404771531015625583641, −5.96041207765668975857242271318, −5.11531299340591306921817263913, −3.12753470840504086292556823109, −2.24569126615295195335036617102, −1.94893835393557463149289811122, −0.03242826700082158098937539989,
0.03242826700082158098937539989, 1.94893835393557463149289811122, 2.24569126615295195335036617102, 3.12753470840504086292556823109, 5.11531299340591306921817263913, 5.96041207765668975857242271318, 6.58656473404771531015625583641, 7.40405564408688640071715591227, 8.010395274096552704671255803635, 8.944322378034296914670106617319
