| L(s) = 1 | − 2·2-s + 4·4-s − 5·5-s − 24.6·7-s − 8·8-s + 10·10-s + 17.3·11-s + 4.00·13-s + 49.2·14-s + 16·16-s − 48.2·17-s + 79.3·19-s − 20·20-s − 34.7·22-s + 23·23-s + 25·25-s − 8.01·26-s − 98.5·28-s + 254.·29-s − 220.·31-s − 32·32-s + 96.5·34-s + 123.·35-s − 422.·37-s − 158.·38-s + 40·40-s + 170.·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s − 1.33·7-s − 0.353·8-s + 0.316·10-s + 0.476·11-s + 0.0854·13-s + 0.940·14-s + 0.250·16-s − 0.688·17-s + 0.957·19-s − 0.223·20-s − 0.336·22-s + 0.208·23-s + 0.200·25-s − 0.0604·26-s − 0.665·28-s + 1.62·29-s − 1.27·31-s − 0.176·32-s + 0.486·34-s + 0.595·35-s − 1.87·37-s − 0.677·38-s + 0.158·40-s + 0.648·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2070 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.7110447438\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7110447438\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 + 24.6T + 343T^{2} \) |
| 11 | \( 1 - 17.3T + 1.33e3T^{2} \) |
| 13 | \( 1 - 4.00T + 2.19e3T^{2} \) |
| 17 | \( 1 + 48.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 79.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 254.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 220.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 422.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 170.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 228.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 580.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 260.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 353.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 80.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 820.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 614.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 511.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 160.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 32.5T + 5.71e5T^{2} \) |
| 89 | \( 1 - 25.0T + 7.04e5T^{2} \) |
| 97 | \( 1 + 249.T + 9.12e5T^{2} \) |
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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
−8.904117056557845273032828129860, −8.093676741718314502517263715601, −7.05391851979683225675462407860, −6.72683194032225287774755082423, −5.82050224959688413755882208105, −4.70408736513141038608060254869, −3.51717466694518102706416589365, −2.99783461432601752691620050922, −1.63851891644538925815566226347, −0.42819437250246830780020400942,
0.42819437250246830780020400942, 1.63851891644538925815566226347, 2.99783461432601752691620050922, 3.51717466694518102706416589365, 4.70408736513141038608060254869, 5.82050224959688413755882208105, 6.72683194032225287774755082423, 7.05391851979683225675462407860, 8.093676741718314502517263715601, 8.904117056557845273032828129860
