| L(s) = 1 | − 4.12·2-s − 494.·4-s − 151.·5-s + 9.40e3·7-s + 4.15e3·8-s + 625.·10-s + 5.69e4·11-s − 6.08e4·13-s − 3.88e4·14-s + 2.36e5·16-s − 8.35e4·17-s + 1.00e6·19-s + 7.50e4·20-s − 2.35e5·22-s − 1.35e6·23-s − 1.93e6·25-s + 2.51e5·26-s − 4.65e6·28-s − 3.12e6·29-s + 2.97e6·31-s − 3.10e6·32-s + 3.44e5·34-s − 1.42e6·35-s + 6.81e5·37-s − 4.17e6·38-s − 6.30e5·40-s + 4.09e6·41-s + ⋯ |
| L(s) = 1 | − 0.182·2-s − 0.966·4-s − 0.108·5-s + 1.48·7-s + 0.358·8-s + 0.0197·10-s + 1.17·11-s − 0.591·13-s − 0.270·14-s + 0.901·16-s − 0.242·17-s + 1.77·19-s + 0.104·20-s − 0.214·22-s − 1.01·23-s − 0.988·25-s + 0.107·26-s − 1.43·28-s − 0.820·29-s + 0.578·31-s − 0.523·32-s + 0.0442·34-s − 0.160·35-s + 0.0597·37-s − 0.324·38-s − 0.0389·40-s + 0.226·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.912646846\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.912646846\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 + 8.35e4T \) |
| good | 2 | \( 1 + 4.12T + 512T^{2} \) |
| 5 | \( 1 + 151.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 9.40e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.69e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 6.08e4T + 1.06e10T^{2} \) |
| 19 | \( 1 - 1.00e6T + 3.22e11T^{2} \) |
| 23 | \( 1 + 1.35e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 3.12e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.97e6T + 2.64e13T^{2} \) |
| 37 | \( 1 - 6.81e5T + 1.29e14T^{2} \) |
| 41 | \( 1 - 4.09e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.00e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 2.54e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 3.14e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 9.03e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.87e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.32e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 4.18e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 4.80e7T + 5.88e16T^{2} \) |
| 79 | \( 1 - 3.49e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.05e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 2.03e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 1.24e9T + 7.60e17T^{2} \) |
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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
−11.49038080267017724758850108338, −10.03523252200868001185572508940, −9.225752831859212206075585637316, −8.173422215609628965963487327132, −7.40632093245297140426837427332, −5.66470355832304364651190537922, −4.67751155522788043192142096426, −3.75431963409008746186499654945, −1.82279688480640415823749087802, −0.77355834393114266705120208879,
0.77355834393114266705120208879, 1.82279688480640415823749087802, 3.75431963409008746186499654945, 4.67751155522788043192142096426, 5.66470355832304364651190537922, 7.40632093245297140426837427332, 8.173422215609628965963487327132, 9.225752831859212206075585637316, 10.03523252200868001185572508940, 11.49038080267017724758850108338
