| L(s) = 1 | − 26.6·2-s + 9.79·3-s + 198.·4-s − 2.46e3·5-s − 260.·6-s − 4.66e3·7-s + 8.36e3·8-s − 1.95e4·9-s + 6.56e4·10-s − 4.63e4·11-s + 1.93e3·12-s − 1.66e5·13-s + 1.24e5·14-s − 2.41e4·15-s − 3.24e5·16-s − 2.96e5·17-s + 5.21e5·18-s + 9.34e3·19-s − 4.87e5·20-s − 4.57e4·21-s + 1.23e6·22-s + 6.41e5·23-s + 8.19e4·24-s + 4.11e6·25-s + 4.42e6·26-s − 3.84e5·27-s − 9.24e5·28-s + ⋯ |
| L(s) = 1 | − 1.17·2-s + 0.0698·3-s + 0.386·4-s − 1.76·5-s − 0.0822·6-s − 0.734·7-s + 0.722·8-s − 0.995·9-s + 2.07·10-s − 0.954·11-s + 0.0270·12-s − 1.61·13-s + 0.865·14-s − 0.123·15-s − 1.23·16-s − 0.861·17-s + 1.17·18-s + 0.0164·19-s − 0.681·20-s − 0.0512·21-s + 1.12·22-s + 0.478·23-s + 0.0504·24-s + 2.10·25-s + 1.89·26-s − 0.139·27-s − 0.284·28-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
| \(L(5)\) |
\(\approx\) |
\(0.005383767390\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.005383767390\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 26.6T + 512T^{2} \) |
| 3 | \( 1 - 9.79T + 1.96e4T^{2} \) |
| 5 | \( 1 + 2.46e3T + 1.95e6T^{2} \) |
| 7 | \( 1 + 4.66e3T + 4.03e7T^{2} \) |
| 11 | \( 1 + 4.63e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.66e5T + 1.06e10T^{2} \) |
| 17 | \( 1 + 2.96e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 9.34e3T + 3.22e11T^{2} \) |
| 23 | \( 1 - 6.41e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.02e6T + 1.45e13T^{2} \) |
| 31 | \( 1 + 5.91e6T + 2.64e13T^{2} \) |
| 41 | \( 1 - 1.25e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 1.45e7T + 5.02e14T^{2} \) |
| 47 | \( 1 + 1.07e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 5.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 8.37e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.21e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + 2.06e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.50e8T + 4.58e16T^{2} \) |
| 73 | \( 1 + 1.66e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.03e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 4.96e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 8.24e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 8.60e8T + 7.60e17T^{2} \) |
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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
−14.71451714743929051248751392517, −12.92885829092627899794111034997, −11.65086786959222352065039285403, −10.56153592141617299485476779935, −9.130802943942622002921931154331, −8.042670333725036344211933377737, −7.17596138698155537174466976215, −4.69044477169190131166684359303, −2.87429177178791909605224917545, −0.05629969284389750893677091239,
0.05629969284389750893677091239, 2.87429177178791909605224917545, 4.69044477169190131166684359303, 7.17596138698155537174466976215, 8.042670333725036344211933377737, 9.130802943942622002921931154331, 10.56153592141617299485476779935, 11.65086786959222352065039285403, 12.92885829092627899794111034997, 14.71451714743929051248751392517
