| L(s) = 1 | + 8.16·2-s − 75.6·3-s − 61.3·4-s − 56.3·5-s − 617.·6-s + 1.19e3·7-s − 1.54e3·8-s + 3.53e3·9-s − 460.·10-s + 2.63e3·11-s + 4.63e3·12-s + 4.97e3·13-s + 9.78e3·14-s + 4.26e3·15-s − 4.77e3·16-s + 1.53e4·17-s + 2.88e4·18-s − 1.24e4·19-s + 3.45e3·20-s − 9.06e4·21-s + 2.14e4·22-s − 7.34e3·23-s + 1.16e5·24-s − 7.49e4·25-s + 4.06e4·26-s − 1.02e5·27-s − 7.34e4·28-s + ⋯ |
| L(s) = 1 | + 0.721·2-s − 1.61·3-s − 0.478·4-s − 0.201·5-s − 1.16·6-s + 1.32·7-s − 1.06·8-s + 1.61·9-s − 0.145·10-s + 0.596·11-s + 0.774·12-s + 0.627·13-s + 0.953·14-s + 0.326·15-s − 0.291·16-s + 0.758·17-s + 1.16·18-s − 0.415·19-s + 0.0966·20-s − 2.13·21-s + 0.430·22-s − 0.125·23-s + 1.72·24-s − 0.959·25-s + 0.453·26-s − 0.998·27-s − 0.632·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.349269247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.349269247\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 37 | \( 1 + 5.06e4T \) |
| good | 2 | \( 1 - 8.16T + 128T^{2} \) |
| 3 | \( 1 + 75.6T + 2.18e3T^{2} \) |
| 5 | \( 1 + 56.3T + 7.81e4T^{2} \) |
| 7 | \( 1 - 1.19e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 2.63e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 4.97e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 1.53e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 1.24e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 7.34e3T + 3.40e9T^{2} \) |
| 29 | \( 1 - 2.34e5T + 1.72e10T^{2} \) |
| 31 | \( 1 - 2.42e5T + 2.75e10T^{2} \) |
| 41 | \( 1 + 4.03e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 4.29e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 3.95e5T + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.06e6T + 1.17e12T^{2} \) |
| 59 | \( 1 - 2.32e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 1.32e6T + 3.14e12T^{2} \) |
| 67 | \( 1 - 9.97e5T + 6.06e12T^{2} \) |
| 71 | \( 1 + 3.83e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.62e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 4.60e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 8.89e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.44e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 2.82e6T + 8.07e13T^{2} \) |
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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
−14.74155618109992616208902291003, −13.61138184619045379578545626904, −12.07112696187169697434437923639, −11.65873135473587346165237996414, −10.27150199790102560758264396649, −8.335514864336633133847352044283, −6.35209199103986789833611454045, −5.20522183194439880990504952980, −4.20827723092694388620753588808, −0.939844912586268810222203608743,
0.939844912586268810222203608743, 4.20827723092694388620753588808, 5.20522183194439880990504952980, 6.35209199103986789833611454045, 8.335514864336633133847352044283, 10.27150199790102560758264396649, 11.65873135473587346165237996414, 12.07112696187169697434437923639, 13.61138184619045379578545626904, 14.74155618109992616208902291003
