| L(s) = 1 | + 204·2-s − 6.56e3·3-s − 8.94e4·4-s − 1.63e5·5-s − 1.33e6·6-s − 2.08e7·7-s − 4.49e7·8-s + 4.30e7·9-s − 3.33e7·10-s + 8.17e8·11-s + 5.86e8·12-s + 2.99e8·13-s − 4.25e9·14-s + 1.07e9·15-s + 2.54e9·16-s − 4.47e10·17-s + 8.78e9·18-s + 7.87e10·19-s + 1.46e10·20-s + 1.36e11·21-s + 1.66e11·22-s − 7.04e11·23-s + 2.95e11·24-s − 7.36e11·25-s + 6.11e10·26-s − 2.82e11·27-s + 1.86e12·28-s + ⋯ |
| L(s) = 1 | + 0.563·2-s − 0.577·3-s − 0.682·4-s − 0.187·5-s − 0.325·6-s − 1.36·7-s − 0.948·8-s + 1/3·9-s − 0.105·10-s + 1.14·11-s + 0.394·12-s + 0.101·13-s − 0.770·14-s + 0.108·15-s + 0.148·16-s − 1.55·17-s + 0.187·18-s + 1.06·19-s + 0.127·20-s + 0.789·21-s + 0.647·22-s − 1.87·23-s + 0.547·24-s − 0.964·25-s + 0.0573·26-s − 0.192·27-s + 0.932·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{8} T \) |
| good | 2 | \( 1 - 51 p^{2} T + p^{17} T^{2} \) |
| 5 | \( 1 + 163554 T + p^{17} T^{2} \) |
| 7 | \( 1 + 425440 p^{2} T + p^{17} T^{2} \) |
| 11 | \( 1 - 817372356 T + p^{17} T^{2} \) |
| 13 | \( 1 - 23045366 p T + p^{17} T^{2} \) |
| 17 | \( 1 + 44775606078 T + p^{17} T^{2} \) |
| 19 | \( 1 - 78748651964 T + p^{17} T^{2} \) |
| 23 | \( 1 + 704672009160 T + p^{17} T^{2} \) |
| 29 | \( 1 + 163793785242 T + p^{17} T^{2} \) |
| 31 | \( 1 - 1049860831400 T + p^{17} T^{2} \) |
| 37 | \( 1 + 19805735857210 T + p^{17} T^{2} \) |
| 41 | \( 1 - 14660035932090 T + p^{17} T^{2} \) |
| 43 | \( 1 - 116038864682564 T + p^{17} T^{2} \) |
| 47 | \( 1 + 176606594594112 T + p^{17} T^{2} \) |
| 53 | \( 1 - 152863496635230 T + p^{17} T^{2} \) |
| 59 | \( 1 + 262797291296124 T + p^{17} T^{2} \) |
| 61 | \( 1 + 1358552281482562 T + p^{17} T^{2} \) |
| 67 | \( 1 - 444863620615292 T + p^{17} T^{2} \) |
| 71 | \( 1 + 4003270764790968 T + p^{17} T^{2} \) |
| 73 | \( 1 - 924832535317130 T + p^{17} T^{2} \) |
| 79 | \( 1 - 14747307742797080 T + p^{17} T^{2} \) |
| 83 | \( 1 - 26422963268810172 T + p^{17} T^{2} \) |
| 89 | \( 1 + 38883748080645126 T + p^{17} T^{2} \) |
| 97 | \( 1 + 25374394856250238 T + p^{17} T^{2} \) |
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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
−22.11637051515949690110736137997, −19.60628610543237216145222594774, −17.78540312069862063897593941383, −15.88585877867854913681454982756, −13.67662376854948998545380433369, −12.09486065933878575593240674694, −9.473710539412500487025916307962, −6.22021921209429111708346581427, −3.92881767815863618531593253568, 0,
3.92881767815863618531593253568, 6.22021921209429111708346581427, 9.473710539412500487025916307962, 12.09486065933878575593240674694, 13.67662376854948998545380433369, 15.88585877867854913681454982756, 17.78540312069862063897593941383, 19.60628610543237216145222594774, 22.11637051515949690110736137997
