| L(s) = 1 | + 4.40·2-s + 198.·3-s − 492.·4-s + 2.55e3·5-s + 873.·6-s + 499.·7-s − 4.42e3·8-s + 1.96e4·9-s + 1.12e4·10-s + 6.66e4·11-s − 9.77e4·12-s − 1.01e5·13-s + 2.19e3·14-s + 5.07e5·15-s + 2.32e5·16-s + 8.39e4·17-s + 8.65e4·18-s − 2.60e5·19-s − 1.25e6·20-s + 9.90e4·21-s + 2.93e5·22-s − 2.12e5·23-s − 8.77e5·24-s + 4.58e6·25-s − 4.48e5·26-s − 5.22e3·27-s − 2.45e5·28-s + ⋯ |
| L(s) = 1 | + 0.194·2-s + 1.41·3-s − 0.962·4-s + 1.83·5-s + 0.275·6-s + 0.0786·7-s − 0.382·8-s + 0.998·9-s + 0.356·10-s + 1.37·11-s − 1.36·12-s − 0.989·13-s + 0.0153·14-s + 2.58·15-s + 0.887·16-s + 0.243·17-s + 0.194·18-s − 0.458·19-s − 1.76·20-s + 0.111·21-s + 0.267·22-s − 0.158·23-s − 0.540·24-s + 2.34·25-s − 0.192·26-s − 0.00189·27-s − 0.0756·28-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
| \(L(5)\) |
\(\approx\) |
\(3.733101597\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.733101597\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 4.40T + 512T^{2} \) |
| 3 | \( 1 - 198.T + 1.96e4T^{2} \) |
| 5 | \( 1 - 2.55e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 499.T + 4.03e7T^{2} \) |
| 11 | \( 1 - 6.66e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + 1.01e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 8.39e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 2.60e5T + 3.22e11T^{2} \) |
| 23 | \( 1 + 2.12e5T + 1.80e12T^{2} \) |
| 29 | \( 1 - 4.96e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 5.82e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + 7.82e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 1.45e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.98e7T + 1.11e15T^{2} \) |
| 53 | \( 1 + 6.68e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 8.48e7T + 8.66e15T^{2} \) |
| 61 | \( 1 + 1.65e8T + 1.16e16T^{2} \) |
| 67 | \( 1 - 2.49e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 3.60e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 3.87e8T + 5.88e16T^{2} \) |
| 79 | \( 1 + 3.48e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 8.29e7T + 1.86e17T^{2} \) |
| 89 | \( 1 - 1.02e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.41e9T + 7.60e17T^{2} \) |
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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
−14.18666932156533456417204187925, −13.64767278977963757348134466431, −12.48655311150705631562658029271, −9.911313148502781157130727033038, −9.412077504941603739496070990698, −8.390308642757234140158136015877, −6.35753682045608809890562152321, −4.68545751603896526402906414865, −2.94914513680485534749720131856, −1.53981476555097069235957262338,
1.53981476555097069235957262338, 2.94914513680485534749720131856, 4.68545751603896526402906414865, 6.35753682045608809890562152321, 8.390308642757234140158136015877, 9.412077504941603739496070990698, 9.911313148502781157130727033038, 12.48655311150705631562658029271, 13.64767278977963757348134466431, 14.18666932156533456417204187925
