| L(s) = 1 | + 528·2-s + 1.47e5·4-s + 1.02e6·5-s + 3.22e6·7-s + 8.78e6·8-s + 5.41e8·10-s + 7.53e8·11-s + 2.54e9·13-s + 1.70e9·14-s − 1.47e10·16-s + 5.42e9·17-s + 1.48e9·19-s + 1.51e11·20-s + 3.97e11·22-s + 3.17e11·23-s + 2.89e11·25-s + 1.34e12·26-s + 4.76e11·28-s − 2.43e12·29-s − 8.84e12·31-s − 8.92e12·32-s + 2.86e12·34-s + 3.30e12·35-s + 1.26e13·37-s + 7.85e11·38-s + 9.01e12·40-s − 4.88e13·41-s + ⋯ |
| L(s) = 1 | + 1.45·2-s + 1.12·4-s + 1.17·5-s + 0.211·7-s + 0.185·8-s + 1.71·10-s + 1.06·11-s + 0.863·13-s + 0.308·14-s − 0.856·16-s + 0.188·17-s + 0.0200·19-s + 1.32·20-s + 1.54·22-s + 0.844·23-s + 0.379·25-s + 1.26·26-s + 0.238·28-s − 0.903·29-s − 1.86·31-s − 1.43·32-s + 0.275·34-s + 0.248·35-s + 0.594·37-s + 0.0293·38-s + 0.217·40-s − 0.955·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(4.920558535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.920558535\) |
| \(L(\frac{19}{2})\) |
|
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 | \( 1 - 33 p^{4} T + p^{17} T^{2} \) |
| 5 | \( 1 - 41034 p^{2} T + p^{17} T^{2} \) |
| 7 | \( 1 - 460856 p T + p^{17} T^{2} \) |
| 11 | \( 1 - 68510748 p T + p^{17} T^{2} \) |
| 13 | \( 1 - 195466502 p T + p^{17} T^{2} \) |
| 17 | \( 1 - 5429742318 T + p^{17} T^{2} \) |
| 19 | \( 1 - 1487499860 T + p^{17} T^{2} \) |
| 23 | \( 1 - 317091823464 T + p^{17} T^{2} \) |
| 29 | \( 1 + 2433410602590 T + p^{17} T^{2} \) |
| 31 | \( 1 + 8849722053088 T + p^{17} T^{2} \) |
| 37 | \( 1 - 12691652946662 T + p^{17} T^{2} \) |
| 41 | \( 1 + 48864151002282 T + p^{17} T^{2} \) |
| 43 | \( 1 + 91019974317844 T + p^{17} T^{2} \) |
| 47 | \( 1 - 49304994276048 T + p^{17} T^{2} \) |
| 53 | \( 1 + 22940453195766 T + p^{17} T^{2} \) |
| 59 | \( 1 + 32695090729980 T + p^{17} T^{2} \) |
| 61 | \( 1 + 1308285854869378 T + p^{17} T^{2} \) |
| 67 | \( 1 - 5196143861984132 T + p^{17} T^{2} \) |
| 71 | \( 1 - 3709489877412408 T + p^{17} T^{2} \) |
| 73 | \( 1 - 3402372968272586 T + p^{17} T^{2} \) |
| 79 | \( 1 - 2366533941308240 T + p^{17} T^{2} \) |
| 83 | \( 1 - 29766750443172204 T + p^{17} T^{2} \) |
| 89 | \( 1 + 29167184100574170 T + p^{17} T^{2} \) |
| 97 | \( 1 + 63769879140957598 T + p^{17} T^{2} \) |
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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
−16.77508903298903502467943815468, −14.93895327333169879198496814549, −13.90802098721009543479243805038, −12.89532141811436053426990751206, −11.27963697813447290393593461438, −9.242444011956127613746661030705, −6.50849372740301783571787090294, −5.32890132848200823228565422142, −3.59845844504404825859105459944, −1.74344336655962927403870215668,
1.74344336655962927403870215668, 3.59845844504404825859105459944, 5.32890132848200823228565422142, 6.50849372740301783571787090294, 9.242444011956127613746661030705, 11.27963697813447290393593461438, 12.89532141811436053426990751206, 13.90802098721009543479243805038, 14.93895327333169879198496814549, 16.77508903298903502467943815468
