| L(s) = 1 | − 29.6·2-s − 44.0·3-s + 364.·4-s + 2.46e3·5-s + 1.30e3·6-s + 1.92e3·7-s + 4.36e3·8-s − 1.77e4·9-s − 7.28e4·10-s − 5.84e3·11-s − 1.60e4·12-s + 6.25e4·13-s − 5.68e4·14-s − 1.08e5·15-s − 3.15e5·16-s − 3.56e5·17-s + 5.25e5·18-s + 8.73e5·19-s + 8.97e5·20-s − 8.45e4·21-s + 1.73e5·22-s − 8.92e5·23-s − 1.92e5·24-s + 4.10e6·25-s − 1.85e6·26-s + 1.64e6·27-s + 7.00e5·28-s + ⋯ |
| L(s) = 1 | − 1.30·2-s − 0.313·3-s + 0.712·4-s + 1.76·5-s + 0.410·6-s + 0.302·7-s + 0.376·8-s − 0.901·9-s − 2.30·10-s − 0.120·11-s − 0.223·12-s + 0.607·13-s − 0.395·14-s − 0.553·15-s − 1.20·16-s − 1.03·17-s + 1.17·18-s + 1.53·19-s + 1.25·20-s − 0.0948·21-s + 0.157·22-s − 0.664·23-s − 0.118·24-s + 2.10·25-s − 0.794·26-s + 0.596·27-s + 0.215·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(1.120917103\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.120917103\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 - 1.87e6T \) |
| good | 2 | \( 1 + 29.6T + 512T^{2} \) |
| 3 | \( 1 + 44.0T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.46e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.92e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 5.84e3T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.25e4T + 1.06e10T^{2} \) |
| 17 | \( 1 + 3.56e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.73e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 8.92e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 2.71e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 2.59e6T + 2.64e13T^{2} \) |
| 41 | \( 1 - 1.54e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 2.89e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 2.29e7T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.34e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.73e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 7.10e6T + 1.16e16T^{2} \) |
| 67 | \( 1 - 1.74e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 8.86e7T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.04e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.75e7T + 1.19e17T^{2} \) |
| 83 | \( 1 - 2.15e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.31e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.73e8T + 7.60e17T^{2} \) |
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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
−14.13153014115328523317463223057, −13.42197472131374852461089733182, −11.44047889963633653702523871111, −10.33018700190686157190277529711, −9.377823911947102564988862800616, −8.362347505871711860360928965118, −6.54684634433179596074670458909, −5.27231014478148648207319424838, −2.30227876282890143192407813051, −0.949399917455686175398581821891,
0.949399917455686175398581821891, 2.30227876282890143192407813051, 5.27231014478148648207319424838, 6.54684634433179596074670458909, 8.362347505871711860360928965118, 9.377823911947102564988862800616, 10.33018700190686157190277529711, 11.44047889963633653702523871111, 13.42197472131374852461089733182, 14.13153014115328523317463223057
