| L(s) = 1 | − 21.1·2-s + 34.2·3-s + 318.·4-s + 463.·5-s − 724.·6-s + 1.27e3·7-s − 4.03e3·8-s − 1.01e3·9-s − 9.80e3·10-s + 5.75e3·11-s + 1.09e4·12-s − 6.79e3·13-s − 2.70e4·14-s + 1.59e4·15-s + 4.44e4·16-s + 2.05e4·17-s + 2.13e4·18-s − 1.48e4·19-s + 1.47e5·20-s + 4.38e4·21-s − 1.21e5·22-s − 4.58e4·23-s − 1.38e5·24-s + 1.36e5·25-s + 1.43e5·26-s − 1.09e5·27-s + 4.07e5·28-s + ⋯ |
| L(s) = 1 | − 1.86·2-s + 0.733·3-s + 2.49·4-s + 1.65·5-s − 1.37·6-s + 1.40·7-s − 2.78·8-s − 0.462·9-s − 3.09·10-s + 1.30·11-s + 1.82·12-s − 0.857·13-s − 2.63·14-s + 1.21·15-s + 2.71·16-s + 1.01·17-s + 0.863·18-s − 0.497·19-s + 4.13·20-s + 1.03·21-s − 2.43·22-s − 0.785·23-s − 2.04·24-s + 1.75·25-s + 1.60·26-s − 1.07·27-s + 3.50·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.468774709\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.468774709\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 5.06e4T \) |
| good | 2 | \( 1 + 21.1T + 128T^{2} \) |
| 3 | \( 1 - 34.2T + 2.18e3T^{2} \) |
| 5 | \( 1 - 463.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.27e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.75e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 6.79e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 2.05e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.48e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 4.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 9.06e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.54e5T + 2.75e10T^{2} \) |
| 41 | \( 1 - 2.97e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.88e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 5.10e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 2.71e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 1.25e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 5.36e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.35e6T + 6.06e12T^{2} \) |
| 71 | \( 1 + 2.17e5T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.42e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 2.42e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.00e7T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.99e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 8.35e6T + 8.07e13T^{2} \) |
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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
−14.63930440598573704698345807585, −14.26833618362755403356012704461, −11.95356190087698862665858852315, −10.62717274278499548504619708220, −9.485419252962689877356124306091, −8.784295452527552887610052232964, −7.55099130177192169395629978869, −5.90206218961254633370570007079, −2.31241121641938364952898666074, −1.42298504082016234708605965643,
1.42298504082016234708605965643, 2.31241121641938364952898666074, 5.90206218961254633370570007079, 7.55099130177192169395629978869, 8.784295452527552887610052232964, 9.485419252962689877356124306091, 10.62717274278499548504619708220, 11.95356190087698862665858852315, 14.26833618362755403356012704461, 14.63930440598573704698345807585
