| L(s) = 1 | − 204·2-s − 8.94e4·4-s + 1.63e5·5-s − 2.08e7·7-s + 4.49e7·8-s − 3.33e7·10-s − 8.17e8·11-s + 2.99e8·13-s + 4.25e9·14-s + 2.54e9·16-s + 4.47e10·17-s + 7.87e10·19-s − 1.46e10·20-s + 1.66e11·22-s + 7.04e11·23-s − 7.36e11·25-s − 6.11e10·26-s + 1.86e12·28-s + 1.63e11·29-s + 1.04e12·31-s − 6.41e12·32-s − 9.13e12·34-s − 3.40e12·35-s − 1.98e13·37-s − 1.60e13·38-s + 7.35e12·40-s − 1.46e13·41-s + ⋯ |
| L(s) = 1 | − 0.563·2-s − 0.682·4-s + 0.187·5-s − 1.36·7-s + 0.948·8-s − 0.105·10-s − 1.14·11-s + 0.101·13-s + 0.770·14-s + 0.148·16-s + 1.55·17-s + 1.06·19-s − 0.127·20-s + 0.647·22-s + 1.87·23-s − 0.964·25-s − 0.0573·26-s + 0.932·28-s + 0.0608·29-s + 0.221·31-s − 1.03·32-s − 0.877·34-s − 0.255·35-s − 0.926·37-s − 0.599·38-s + 0.177·40-s − 0.286·41-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
| \(L(9)\) |
\(\approx\) |
\(0.8664842743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8664842743\) |
| \(L(\frac{19}{2})\) |
|
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 | \( 1 + 51 p^{2} T + p^{17} T^{2} \) |
| 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
−17.05713298292458662381623942753, −15.82014975695620794717103485743, −13.79001667683877738717027114581, −12.68011904032940749123043756500, −10.31600412771330461315078578443, −9.308322441544142239184673955290, −7.55042133894508684350884916741, −5.41906333919262865569931235526, −3.22912613322236515520765374465, −0.73482617135862161810399658197,
0.73482617135862161810399658197, 3.22912613322236515520765374465, 5.41906333919262865569931235526, 7.55042133894508684350884916741, 9.308322441544142239184673955290, 10.31600412771330461315078578443, 12.68011904032940749123043756500, 13.79001667683877738717027114581, 15.82014975695620794717103485743, 17.05713298292458662381623942753
