| L(s) = 1 | + 2·2-s + 4·4-s − 8.84·5-s − 32.7·7-s + 8·8-s − 17.6·10-s + 32.8·11-s − 64.8·13-s − 65.4·14-s + 16·16-s + 88.7·17-s − 112.·19-s − 35.3·20-s + 65.7·22-s − 71.7·23-s − 46.8·25-s − 129.·26-s − 130.·28-s + 209.·29-s − 9.65·31-s + 32·32-s + 177.·34-s + 289.·35-s − 117.·37-s − 224.·38-s − 70.7·40-s + 8.83·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.790·5-s − 1.76·7-s + 0.353·8-s − 0.559·10-s + 0.901·11-s − 1.38·13-s − 1.24·14-s + 0.250·16-s + 1.26·17-s − 1.35·19-s − 0.395·20-s + 0.637·22-s − 0.650·23-s − 0.374·25-s − 0.978·26-s − 0.882·28-s + 1.34·29-s − 0.0559·31-s + 0.176·32-s + 0.895·34-s + 1.39·35-s − 0.522·37-s − 0.957·38-s − 0.279·40-s + 0.0336·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1458 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.624503140\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.624503140\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 8.84T + 125T^{2} \) |
| 7 | \( 1 + 32.7T + 343T^{2} \) |
| 11 | \( 1 - 32.8T + 1.33e3T^{2} \) |
| 13 | \( 1 + 64.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 88.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 112.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 71.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 209.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 9.65T + 2.97e4T^{2} \) |
| 37 | \( 1 + 117.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 8.83T + 6.89e4T^{2} \) |
| 43 | \( 1 - 205.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 188.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 52.1T + 1.48e5T^{2} \) |
| 59 | \( 1 + 270.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 229.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 212.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 336.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.17e3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 55.5T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.15e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.30e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.47e3T + 9.12e5T^{2} \) |
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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
−9.332992709913389429746064296658, −8.191863552625070845630510180386, −7.34241106376531662127579828839, −6.57488129567576889938081089697, −6.03114312792882488943242856731, −4.81805827617733958118493819692, −3.88804482891049809522097073459, −3.31330103108904820327519886821, −2.27897013844265253408552863847, −0.53443182451162067543992986803,
0.53443182451162067543992986803, 2.27897013844265253408552863847, 3.31330103108904820327519886821, 3.88804482891049809522097073459, 4.81805827617733958118493819692, 6.03114312792882488943242856731, 6.57488129567576889938081089697, 7.34241106376531662127579828839, 8.191863552625070845630510180386, 9.332992709913389429746064296658
