| L(s) = 1 | − 7.40e10·2-s − 3.23e22·4-s − 2.38e26·5-s − 3.39e31·7-s + 5.18e33·8-s + 1.76e37·10-s − 8.78e37·11-s + 6.64e41·13-s + 2.51e42·14-s + 8.36e44·16-s + 2.18e46·17-s + 1.29e48·19-s + 7.71e48·20-s + 6.50e48·22-s + 3.32e50·23-s + 3.06e52·25-s − 4.91e52·26-s + 1.09e54·28-s + 4.29e54·29-s + 3.65e55·31-s − 2.57e56·32-s − 1.61e57·34-s + 8.10e57·35-s + 1.09e59·37-s − 9.57e58·38-s − 1.23e60·40-s + 3.98e60·41-s + ⋯ |
| L(s) = 1 | − 0.380·2-s − 0.855·4-s − 1.46·5-s − 0.690·7-s + 0.706·8-s + 0.559·10-s − 0.0778·11-s + 1.12·13-s + 0.263·14-s + 0.586·16-s + 1.57·17-s + 1.44·19-s + 1.25·20-s + 0.0296·22-s + 0.286·23-s + 1.15·25-s − 0.426·26-s + 0.590·28-s + 0.621·29-s + 0.433·31-s − 0.929·32-s − 0.599·34-s + 1.01·35-s + 1.70·37-s − 0.548·38-s − 1.03·40-s + 1.32·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(76-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+75/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(38)\) |
\(\approx\) |
\(1.417516806\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.417516806\) |
| \(L(\frac{77}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + 7.40e10T + 3.77e22T^{2} \) |
| 5 | \( 1 + 2.38e26T + 2.64e52T^{2} \) |
| 7 | \( 1 + 3.39e31T + 2.41e63T^{2} \) |
| 11 | \( 1 + 8.78e37T + 1.27e78T^{2} \) |
| 13 | \( 1 - 6.64e41T + 3.51e83T^{2} \) |
| 17 | \( 1 - 2.18e46T + 1.92e92T^{2} \) |
| 19 | \( 1 - 1.29e48T + 8.06e95T^{2} \) |
| 23 | \( 1 - 3.32e50T + 1.34e102T^{2} \) |
| 29 | \( 1 - 4.29e54T + 4.78e109T^{2} \) |
| 31 | \( 1 - 3.65e55T + 7.11e111T^{2} \) |
| 37 | \( 1 - 1.09e59T + 4.12e117T^{2} \) |
| 41 | \( 1 - 3.98e60T + 9.09e120T^{2} \) |
| 43 | \( 1 + 9.63e60T + 3.23e122T^{2} \) |
| 47 | \( 1 - 4.30e62T + 2.55e125T^{2} \) |
| 53 | \( 1 + 2.27e64T + 2.09e129T^{2} \) |
| 59 | \( 1 + 9.13e64T + 6.51e132T^{2} \) |
| 61 | \( 1 - 1.11e67T + 7.93e133T^{2} \) |
| 67 | \( 1 + 1.68e68T + 9.02e136T^{2} \) |
| 71 | \( 1 + 4.77e69T + 6.98e138T^{2} \) |
| 73 | \( 1 - 2.83e69T + 5.61e139T^{2} \) |
| 79 | \( 1 - 9.67e70T + 2.09e142T^{2} \) |
| 83 | \( 1 - 1.78e72T + 8.52e143T^{2} \) |
| 89 | \( 1 - 1.98e73T + 1.60e146T^{2} \) |
| 97 | \( 1 - 7.38e73T + 1.01e149T^{2} \) |
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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
−10.14682754221793733318637354746, −9.166426674429820410649189713445, −8.049146168092254377692263554654, −7.50397996813812578585193623345, −5.95530924275436244971780190340, −4.71890533001172061758774154221, −3.68612317738190651036538608277, −3.14880026744735482325804755066, −0.986383830161287804296395402040, −0.68496628529995557091908096570,
0.68496628529995557091908096570, 0.986383830161287804296395402040, 3.14880026744735482325804755066, 3.68612317738190651036538608277, 4.71890533001172061758774154221, 5.95530924275436244971780190340, 7.50397996813812578585193623345, 8.049146168092254377692263554654, 9.166426674429820410649189713445, 10.14682754221793733318637354746
