| L(s) = 1 | + 19.7·2-s − 4.59·3-s − 123.·4-s + 1.29e3·5-s − 90.6·6-s + 1.09e4·7-s − 1.25e4·8-s − 1.96e4·9-s + 2.55e4·10-s − 1.21e4·11-s + 565.·12-s + 1.31e5·13-s + 2.15e5·14-s − 5.94e3·15-s − 1.84e5·16-s + 5.59e5·17-s − 3.87e5·18-s + 8.34e5·19-s − 1.59e5·20-s − 5.02e4·21-s − 2.38e5·22-s + 1.31e6·23-s + 5.75e4·24-s − 2.77e5·25-s + 2.58e6·26-s + 1.80e5·27-s − 1.34e6·28-s + ⋯ |
| L(s) = 1 | + 0.871·2-s − 0.0327·3-s − 0.240·4-s + 0.926·5-s − 0.0285·6-s + 1.72·7-s − 1.08·8-s − 0.998·9-s + 0.807·10-s − 0.249·11-s + 0.00786·12-s + 1.27·13-s + 1.50·14-s − 0.0303·15-s − 0.701·16-s + 1.62·17-s − 0.870·18-s + 1.46·19-s − 0.222·20-s − 0.0563·21-s − 0.217·22-s + 0.976·23-s + 0.0354·24-s − 0.141·25-s + 1.10·26-s + 0.0654·27-s − 0.413·28-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
| \(L(5)\) |
\(\approx\) |
\(3.428502205\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.428502205\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 - 19.7T + 512T^{2} \) |
| 3 | \( 1 + 4.59T + 1.96e4T^{2} \) |
| 5 | \( 1 - 1.29e3T + 1.95e6T^{2} \) |
| 7 | \( 1 - 1.09e4T + 4.03e7T^{2} \) |
| 11 | \( 1 + 1.21e4T + 2.35e9T^{2} \) |
| 13 | \( 1 - 1.31e5T + 1.06e10T^{2} \) |
| 17 | \( 1 - 5.59e5T + 1.18e11T^{2} \) |
| 19 | \( 1 - 8.34e5T + 3.22e11T^{2} \) |
| 23 | \( 1 - 1.31e6T + 1.80e12T^{2} \) |
| 29 | \( 1 - 8.68e5T + 1.45e13T^{2} \) |
| 31 | \( 1 + 7.91e6T + 2.64e13T^{2} \) |
| 41 | \( 1 + 1.44e7T + 3.27e14T^{2} \) |
| 43 | \( 1 + 9.44e6T + 5.02e14T^{2} \) |
| 47 | \( 1 + 5.75e5T + 1.11e15T^{2} \) |
| 53 | \( 1 + 2.25e7T + 3.29e15T^{2} \) |
| 59 | \( 1 + 1.47e8T + 8.66e15T^{2} \) |
| 61 | \( 1 + 2.95e7T + 1.16e16T^{2} \) |
| 67 | \( 1 + 3.13e8T + 2.72e16T^{2} \) |
| 71 | \( 1 - 1.99e8T + 4.58e16T^{2} \) |
| 73 | \( 1 - 1.66e8T + 5.88e16T^{2} \) |
| 79 | \( 1 - 1.16e8T + 1.19e17T^{2} \) |
| 83 | \( 1 - 6.63e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + 7.14e8T + 3.50e17T^{2} \) |
| 97 | \( 1 + 1.19e9T + 7.60e17T^{2} \) |
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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
−14.11611922737116457389556941576, −13.61471681468779362985655048240, −12.02347958949877641651012440321, −10.98023944125637654819442642879, −9.240023691424352791670609049850, −8.026654911720482923430411921191, −5.71161974371556351839973932457, −5.16717811331044113834784356257, −3.26776566920780905089894999147, −1.34365803971355645964773180573,
1.34365803971355645964773180573, 3.26776566920780905089894999147, 5.16717811331044113834784356257, 5.71161974371556351839973932457, 8.026654911720482923430411921191, 9.240023691424352791670609049850, 10.98023944125637654819442642879, 12.02347958949877641651012440321, 13.61471681468779362985655048240, 14.11611922737116457389556941576
