| L(s) = 1 | + 38.9·2-s − 235.·3-s + 1.00e3·4-s + 774.·5-s − 9.19e3·6-s − 1.44e3·7-s + 1.93e4·8-s + 3.59e4·9-s + 3.01e4·10-s + 6.32e4·11-s − 2.37e5·12-s + 6.25e4·13-s − 5.64e4·14-s − 1.82e5·15-s + 2.37e5·16-s + 3.88e5·17-s + 1.40e6·18-s + 5.09e5·19-s + 7.80e5·20-s + 3.41e5·21-s + 2.46e6·22-s + 9.73e5·23-s − 4.55e6·24-s − 1.35e6·25-s + 2.43e6·26-s − 3.84e6·27-s − 1.45e6·28-s + ⋯ |
| L(s) = 1 | + 1.72·2-s − 1.68·3-s + 1.96·4-s + 0.554·5-s − 2.89·6-s − 0.227·7-s + 1.66·8-s + 1.82·9-s + 0.954·10-s + 1.30·11-s − 3.30·12-s + 0.607·13-s − 0.392·14-s − 0.932·15-s + 0.904·16-s + 1.12·17-s + 3.15·18-s + 0.896·19-s + 1.09·20-s + 0.383·21-s + 2.24·22-s + 0.725·23-s − 2.80·24-s − 0.692·25-s + 1.04·26-s − 1.39·27-s − 0.448·28-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
| \(L(5)\) |
\(\approx\) |
\(3.654713717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.654713717\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 38.9T + 512T^{2} \) |
| 3 | \( 1 + 235.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 774.T + 1.95e6T^{2} \) |
| 7 | \( 1 + 1.44e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.32e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 6.25e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 3.88e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 5.09e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 9.73e5T + 1.80e12T^{2} \) |
| 29 | \( 1 + 4.77e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.43e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + 2.69e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 3.26e7T + 5.02e14T^{2} \) |
| 47 | \( 1 - 5.42e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.21e7T + 3.29e15T^{2} \) |
| 59 | \( 1 - 3.77e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 9.65e7T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.94e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 9.10e6T + 4.58e16T^{2} \) |
| 73 | \( 1 - 8.88e6T + 5.88e16T^{2} \) |
| 79 | \( 1 + 4.87e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 1.42e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 4.01e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 5.68e7T + 7.60e17T^{2} \) |
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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
−14.11257709987900790206048431783, −13.07516684484342240614401753704, −11.92468690776375814763525794672, −11.43176807005928347833789995022, −9.930386415651450441486392127229, −6.84525632632950110346005175960, −5.99043541960771352465683408737, −5.12816702755769201290319198279, −3.66444978540478582522487541431, −1.29153136492935068118564640766,
1.29153136492935068118564640766, 3.66444978540478582522487541431, 5.12816702755769201290319198279, 5.99043541960771352465683408737, 6.84525632632950110346005175960, 9.930386415651450441486392127229, 11.43176807005928347833789995022, 11.92468690776375814763525794672, 13.07516684484342240614401753704, 14.11257709987900790206048431783
