| L(s) = 1 | − 128·2-s − 918·3-s + 1.63e4·4-s − 7.81e4·5-s + 1.17e5·6-s − 9.53e5·7-s − 2.09e6·8-s − 1.35e7·9-s + 1.00e7·10-s + 1.77e7·11-s − 1.50e7·12-s + 1.40e8·13-s + 1.22e8·14-s + 7.17e7·15-s + 2.68e8·16-s + 2.99e9·17-s + 1.72e9·18-s + 3.25e9·19-s − 1.28e9·20-s + 8.75e8·21-s − 2.27e9·22-s + 6.77e9·23-s + 1.92e9·24-s + 6.10e9·25-s − 1.79e10·26-s + 2.55e10·27-s − 1.56e10·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.242·3-s + 1/2·4-s − 0.447·5-s + 0.171·6-s − 0.437·7-s − 0.353·8-s − 0.941·9-s + 0.316·10-s + 0.275·11-s − 0.121·12-s + 0.621·13-s + 0.309·14-s + 0.108·15-s + 1/4·16-s + 1.77·17-s + 0.665·18-s + 0.835·19-s − 0.223·20-s + 0.106·21-s − 0.194·22-s + 0.414·23-s + 0.0856·24-s + 1/5·25-s − 0.439·26-s + 0.470·27-s − 0.218·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(\approx\) |
\(0.9750630959\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9750630959\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + p^{7} T \) |
| 5 | \( 1 + p^{7} T \) |
| good | 3 | \( 1 + 34 p^{3} T + p^{15} T^{2} \) |
| 7 | \( 1 + 136222 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 17783232 T + p^{15} T^{2} \) |
| 13 | \( 1 - 140533322 T + p^{15} T^{2} \) |
| 17 | \( 1 - 2998870746 T + p^{15} T^{2} \) |
| 19 | \( 1 - 3255852500 T + p^{15} T^{2} \) |
| 23 | \( 1 - 6774812202 T + p^{15} T^{2} \) |
| 29 | \( 1 + 7340322690 T + p^{15} T^{2} \) |
| 31 | \( 1 + 115428411388 T + p^{15} T^{2} \) |
| 37 | \( 1 - 150300986906 T + p^{15} T^{2} \) |
| 41 | \( 1 - 1841603525142 T + p^{15} T^{2} \) |
| 43 | \( 1 - 1510018315682 T + p^{15} T^{2} \) |
| 47 | \( 1 - 6093750843366 T + p^{15} T^{2} \) |
| 53 | \( 1 + 8267412829038 T + p^{15} T^{2} \) |
| 59 | \( 1 + 23516883061980 T + p^{15} T^{2} \) |
| 61 | \( 1 + 3135369104278 T + p^{15} T^{2} \) |
| 67 | \( 1 + 36030983954794 T + p^{15} T^{2} \) |
| 71 | \( 1 - 52169735384172 T + p^{15} T^{2} \) |
| 73 | \( 1 - 69977143684082 T + p^{15} T^{2} \) |
| 79 | \( 1 + 135317670906760 T + p^{15} T^{2} \) |
| 83 | \( 1 - 427456158822882 T + p^{15} T^{2} \) |
| 89 | \( 1 + 446581617299190 T + p^{15} T^{2} \) |
| 97 | \( 1 - 181247411845826 T + p^{15} T^{2} \) |
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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
−17.02391021662394761616130811086, −16.01282035796933310215699984777, −14.34421919933836557870858409812, −12.25500517000625797909187486208, −10.97324147676039509568884792879, −9.272993327032065166427897333731, −7.69130267751382162193782257012, −5.84438928986124289792383426273, −3.23117180850074466310018201271, −0.838552670584454164746528718092,
0.838552670584454164746528718092, 3.23117180850074466310018201271, 5.84438928986124289792383426273, 7.69130267751382162193782257012, 9.272993327032065166427897333731, 10.97324147676039509568884792879, 12.25500517000625797909187486208, 14.34421919933836557870858409812, 16.01282035796933310215699984777, 17.02391021662394761616130811086
