| L(s) = 1 | − 2.18e3·3-s − 2.21e5·5-s + 2.14e6·7-s + 4.78e6·9-s − 3.71e7·11-s − 2.79e8·13-s + 4.84e8·15-s + 2.49e9·17-s + 4.66e9·19-s − 4.69e9·21-s + 1.84e10·23-s + 1.85e10·25-s − 1.04e10·27-s − 1.15e11·29-s + 5.61e10·31-s + 8.12e10·33-s − 4.75e11·35-s + 6.14e11·37-s + 6.12e11·39-s + 5.49e11·41-s + 9.82e11·43-s − 1.05e12·45-s − 2.07e12·47-s − 1.29e11·49-s − 5.45e12·51-s − 1.20e13·53-s + 8.23e12·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.26·5-s + 0.986·7-s + 1/3·9-s − 0.575·11-s − 1.23·13-s + 0.732·15-s + 1.47·17-s + 1.19·19-s − 0.569·21-s + 1.13·23-s + 0.607·25-s − 0.192·27-s − 1.24·29-s + 0.366·31-s + 0.332·33-s − 1.25·35-s + 1.06·37-s + 0.714·39-s + 0.440·41-s + 0.551·43-s − 0.422·45-s − 0.597·47-s − 0.0272·49-s − 0.850·51-s − 1.40·53-s + 0.729·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p^{7} T \) |
| good | 5 | \( 1 + 44298 p T + p^{15} T^{2} \) |
| 7 | \( 1 - 307000 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 37169316 T + p^{15} T^{2} \) |
| 13 | \( 1 + 21536482 p T + p^{15} T^{2} \) |
| 17 | \( 1 - 2492912754 T + p^{15} T^{2} \) |
| 19 | \( 1 - 4669782244 T + p^{15} T^{2} \) |
| 23 | \( 1 - 18467933400 T + p^{15} T^{2} \) |
| 29 | \( 1 + 115953449418 T + p^{15} T^{2} \) |
| 31 | \( 1 - 56187023200 T + p^{15} T^{2} \) |
| 37 | \( 1 - 614764926830 T + p^{15} T^{2} \) |
| 41 | \( 1 - 549859792410 T + p^{15} T^{2} \) |
| 43 | \( 1 - 982884444028 T + p^{15} T^{2} \) |
| 47 | \( 1 + 2076144322896 T + p^{15} T^{2} \) |
| 53 | \( 1 + 12048378188130 T + p^{15} T^{2} \) |
| 59 | \( 1 + 23087905758324 T + p^{15} T^{2} \) |
| 61 | \( 1 + 8505809142442 T + p^{15} T^{2} \) |
| 67 | \( 1 - 12331010771476 T + p^{15} T^{2} \) |
| 71 | \( 1 + 58989192692472 T + p^{15} T^{2} \) |
| 73 | \( 1 + 5609828808070 T + p^{15} T^{2} \) |
| 79 | \( 1 + 159918683826800 T + p^{15} T^{2} \) |
| 83 | \( 1 + 57675894342876 T + p^{15} T^{2} \) |
| 89 | \( 1 + 362287610413974 T + p^{15} T^{2} \) |
| 97 | \( 1 + 539786645144926 T + p^{15} T^{2} \) |
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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
−11.76640318987411981125305880103, −11.05136959786273537483944931887, −9.647937253390338741150110056164, −7.82132118894746928648970407893, −7.46409228671406316292639810505, −5.42844263556466885156487243162, −4.55652243411449712015572482681, −3.06558229832775082467873643208, −1.21362746662839115058051787843, 0,
1.21362746662839115058051787843, 3.06558229832775082467873643208, 4.55652243411449712015572482681, 5.42844263556466885156487243162, 7.46409228671406316292639810505, 7.82132118894746928648970407893, 9.647937253390338741150110056164, 11.05136959786273537483944931887, 11.76640318987411981125305880103
