| L(s) = 1 | − 40.4·2-s − 163.·3-s + 1.12e3·4-s + 517.·5-s + 6.62e3·6-s + 1.48e3·7-s − 2.47e4·8-s + 7.17e3·9-s − 2.09e4·10-s − 3.11e4·11-s − 1.84e5·12-s − 1.08e5·13-s − 5.99e4·14-s − 8.47e4·15-s + 4.25e5·16-s + 2.19e5·17-s − 2.90e5·18-s − 7.50e5·19-s + 5.81e5·20-s − 2.43e5·21-s + 1.26e6·22-s + 1.21e6·23-s + 4.05e6·24-s − 1.68e6·25-s + 4.40e6·26-s + 2.04e6·27-s + 1.66e6·28-s + ⋯ |
| L(s) = 1 | − 1.78·2-s − 1.16·3-s + 2.19·4-s + 0.369·5-s + 2.08·6-s + 0.233·7-s − 2.13·8-s + 0.364·9-s − 0.661·10-s − 0.642·11-s − 2.56·12-s − 1.05·13-s − 0.417·14-s − 0.432·15-s + 1.62·16-s + 0.636·17-s − 0.651·18-s − 1.32·19-s + 0.812·20-s − 0.272·21-s + 1.14·22-s + 0.905·23-s + 2.49·24-s − 0.863·25-s + 1.89·26-s + 0.742·27-s + 0.512·28-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
| \(L(5)\) |
\(\approx\) |
\(0.3222018142\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3222018142\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 40.4T + 512T^{2} \) |
| 3 | \( 1 + 163.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 517.T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.48e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 3.11e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.08e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 2.19e5T + 1.18e11T^{2} \) |
| 19 | \( 1 + 7.50e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.21e6T + 1.80e12T^{2} \) |
| 29 | \( 1 + 2.30e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 4.05e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + 3.36e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.84e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 6.81e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 1.57e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 1.90e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.95e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.29e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 3.16e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.71e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 2.72e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 7.19e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 8.51e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.72e9T + 7.60e17T^{2} \) |
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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
−14.84546658054328840034826253801, −12.62244278721773127764122610395, −11.37056597906886690900691101289, −10.55434171558784862723255490337, −9.524601466314586949219350713095, −8.044713773870232468708969156860, −6.77496776217071347663710788658, −5.38858739093451458650415908593, −2.17202539288376808940839761696, −0.52289016750729156185394977502,
0.52289016750729156185394977502, 2.17202539288376808940839761696, 5.38858739093451458650415908593, 6.77496776217071347663710788658, 8.044713773870232468708969156860, 9.524601466314586949219350713095, 10.55434171558784862723255490337, 11.37056597906886690900691101289, 12.62244278721773127764122610395, 14.84546658054328840034826253801
