| L(s) = 1 | − 30.1·2-s + 194.·3-s + 399.·4-s − 15.3·5-s − 5.86e3·6-s + 1.04e4·7-s + 3.40e3·8-s + 1.80e4·9-s + 463.·10-s − 2.30e3·11-s + 7.74e4·12-s − 4.67e4·13-s − 3.15e5·14-s − 2.98e3·15-s − 3.07e5·16-s + 6.19e4·17-s − 5.43e5·18-s − 1.64e5·19-s − 6.12e3·20-s + 2.02e6·21-s + 6.95e4·22-s + 2.50e6·23-s + 6.61e5·24-s − 1.95e6·25-s + 1.41e6·26-s − 3.22e5·27-s + 4.16e6·28-s + ⋯ |
| L(s) = 1 | − 1.33·2-s + 1.38·3-s + 0.779·4-s − 0.0109·5-s − 1.84·6-s + 1.64·7-s + 0.294·8-s + 0.915·9-s + 0.0146·10-s − 0.0474·11-s + 1.07·12-s − 0.454·13-s − 2.19·14-s − 0.0152·15-s − 1.17·16-s + 0.179·17-s − 1.22·18-s − 0.289·19-s − 0.00856·20-s + 2.27·21-s + 0.0633·22-s + 1.86·23-s + 0.407·24-s − 0.999·25-s + 0.605·26-s − 0.116·27-s + 1.28·28-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
| \(L(5)\) |
\(\approx\) |
\(1.803788177\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.803788177\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + 30.1T + 512T^{2} \) |
| 3 | \( 1 - 194.T + 1.96e4T^{2} \) |
| 5 | \( 1 + 15.3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.04e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 2.30e3T + 2.35e9T^{2} \) |
| 13 | \( 1 + 4.67e4T + 1.06e10T^{2} \) |
| 17 | \( 1 - 6.19e4T + 1.18e11T^{2} \) |
| 19 | \( 1 + 1.64e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 2.50e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 3.41e6T + 1.45e13T^{2} \) |
| 31 | \( 1 - 8.37e6T + 2.64e13T^{2} \) |
| 41 | \( 1 - 5.61e6T + 3.27e14T^{2} \) |
| 43 | \( 1 - 4.37e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 3.89e6T + 1.11e15T^{2} \) |
| 53 | \( 1 - 6.33e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.62e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.77e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 1.07e8T + 2.72e16T^{2} \) |
| 71 | \( 1 + 2.09e5T + 4.58e16T^{2} \) |
| 73 | \( 1 + 4.13e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 2.63e8T + 1.19e17T^{2} \) |
| 83 | \( 1 + 5.71e8T + 1.86e17T^{2} \) |
| 89 | \( 1 - 7.10e8T + 3.50e17T^{2} \) |
| 97 | \( 1 - 7.33e8T + 7.60e17T^{2} \) |
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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
−14.49470254074104923980361536486, −13.51797106206856528320254237956, −11.54454949886031368031290443489, −10.26100758038553723511992392084, −8.982358440399197924257130913166, −8.233307982505514709937878200977, −7.41543339120954041707399991035, −4.61338861697821882567432415761, −2.46147595266569085203623529692, −1.19487171889471616669133627368,
1.19487171889471616669133627368, 2.46147595266569085203623529692, 4.61338861697821882567432415761, 7.41543339120954041707399991035, 8.233307982505514709937878200977, 8.982358440399197924257130913166, 10.26100758038553723511992392084, 11.54454949886031368031290443489, 13.51797106206856528320254237956, 14.49470254074104923980361536486
