| L(s) = 1 | + 5.43·2-s + 85.9·3-s − 98.4·4-s − 143.·5-s + 467.·6-s + 1.59e3·7-s − 1.23e3·8-s + 5.20e3·9-s − 782.·10-s + 6.06e3·11-s − 8.46e3·12-s − 1.78e3·13-s + 8.65e3·14-s − 1.23e4·15-s + 5.90e3·16-s − 2.42e4·17-s + 2.82e4·18-s − 9.57e3·19-s + 1.41e4·20-s + 1.36e5·21-s + 3.29e4·22-s + 7.71e4·23-s − 1.05e5·24-s − 5.74e4·25-s − 9.70e3·26-s + 2.58e5·27-s − 1.56e5·28-s + ⋯ |
| L(s) = 1 | + 0.480·2-s + 1.83·3-s − 0.769·4-s − 0.514·5-s + 0.883·6-s + 1.75·7-s − 0.850·8-s + 2.37·9-s − 0.247·10-s + 1.37·11-s − 1.41·12-s − 0.225·13-s + 0.843·14-s − 0.946·15-s + 0.360·16-s − 1.19·17-s + 1.14·18-s − 0.320·19-s + 0.396·20-s + 3.22·21-s + 0.660·22-s + 1.32·23-s − 1.56·24-s − 0.734·25-s − 0.108·26-s + 2.53·27-s − 1.34·28-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
| \(L(4)\) |
\(\approx\) |
\(3.629875971\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.629875971\) |
| \(L(\frac{9}{2})\) |
|
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 | \( 1 - 5.43T + 128T^{2} \) |
| 3 | \( 1 - 85.9T + 2.18e3T^{2} \) |
| 5 | \( 1 + 143.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.59e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 6.06e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.78e3T + 6.27e7T^{2} \) |
| 17 | \( 1 + 2.42e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 9.57e3T + 8.93e8T^{2} \) |
| 23 | \( 1 - 7.71e4T + 3.40e9T^{2} \) |
| 29 | \( 1 + 1.43e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 2.83e5T + 2.75e10T^{2} \) |
| 41 | \( 1 - 5.73e3T + 1.94e11T^{2} \) |
| 43 | \( 1 + 1.06e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 9.30e4T + 5.06e11T^{2} \) |
| 53 | \( 1 + 1.28e6T + 1.17e12T^{2} \) |
| 59 | \( 1 + 5.43e5T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.26e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.77e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 4.49e5T + 9.09e12T^{2} \) |
| 73 | \( 1 - 1.86e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.91e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 1.36e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + 2.42e6T + 4.42e13T^{2} \) |
| 97 | \( 1 - 7.89e6T + 8.07e13T^{2} \) |
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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
−14.71538210988669039358024477270, −13.98515200542707558874581950107, −12.88101821161973296595762569673, −11.33267408955845907612830322463, −9.210799352699597717735040497236, −8.622742097982888532969094597096, −7.43811162983219110489926942212, −4.61903572998725857207479342440, −3.71812872251981761579952682874, −1.75921362187966111280090576783,
1.75921362187966111280090576783, 3.71812872251981761579952682874, 4.61903572998725857207479342440, 7.43811162983219110489926942212, 8.622742097982888532969094597096, 9.210799352699597717735040497236, 11.33267408955845907612830322463, 12.88101821161973296595762569673, 13.98515200542707558874581950107, 14.71538210988669039358024477270
