| L(s) = 1 | + 64·2-s + 1.23e3·3-s + 4.09e3·4-s − 5.74e4·5-s + 7.91e4·6-s + 6.42e4·7-s + 2.62e5·8-s − 6.66e4·9-s − 3.67e6·10-s + 2.46e6·11-s + 5.06e6·12-s + 8.03e6·13-s + 4.11e6·14-s − 7.10e7·15-s + 1.67e7·16-s + 7.11e7·17-s − 4.26e6·18-s + 1.36e8·19-s − 2.35e8·20-s + 7.93e7·21-s + 1.57e8·22-s − 1.18e9·23-s + 3.24e8·24-s + 2.07e9·25-s + 5.14e8·26-s − 2.05e9·27-s + 2.63e8·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.978·3-s + 1/2·4-s − 1.64·5-s + 0.692·6-s + 0.206·7-s + 0.353·8-s − 0.0417·9-s − 1.16·10-s + 0.419·11-s + 0.489·12-s + 0.461·13-s + 0.145·14-s − 1.60·15-s + 1/4·16-s + 0.714·17-s − 0.0295·18-s + 0.664·19-s − 0.822·20-s + 0.201·21-s + 0.296·22-s − 1.67·23-s + 0.346·24-s + 1.70·25-s + 0.326·26-s − 1.01·27-s + 0.103·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.918623084\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.918623084\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{6} T \) |
| good | 3 | \( 1 - 412 p T + p^{13} T^{2} \) |
| 5 | \( 1 + 2298 p^{2} T + p^{13} T^{2} \) |
| 7 | \( 1 - 9176 p T + p^{13} T^{2} \) |
| 11 | \( 1 - 224052 p T + p^{13} T^{2} \) |
| 13 | \( 1 - 8032766 T + p^{13} T^{2} \) |
| 17 | \( 1 - 71112402 T + p^{13} T^{2} \) |
| 19 | \( 1 - 136337060 T + p^{13} T^{2} \) |
| 23 | \( 1 + 1186563144 T + p^{13} T^{2} \) |
| 29 | \( 1 + 890583090 T + p^{13} T^{2} \) |
| 31 | \( 1 - 4595552672 T + p^{13} T^{2} \) |
| 37 | \( 1 + 19585053898 T + p^{13} T^{2} \) |
| 41 | \( 1 + 2724170358 T + p^{13} T^{2} \) |
| 43 | \( 1 - 51762321116 T + p^{13} T^{2} \) |
| 47 | \( 1 + 53572833168 T + p^{13} T^{2} \) |
| 53 | \( 1 - 82633440006 T + p^{13} T^{2} \) |
| 59 | \( 1 + 394266352980 T + p^{13} T^{2} \) |
| 61 | \( 1 - 671061772142 T + p^{13} T^{2} \) |
| 67 | \( 1 - 388156449812 T + p^{13} T^{2} \) |
| 71 | \( 1 + 388772243928 T + p^{13} T^{2} \) |
| 73 | \( 1 - 1540972938026 T + p^{13} T^{2} \) |
| 79 | \( 1 + 3306509559280 T + p^{13} T^{2} \) |
| 83 | \( 1 - 4931756967396 T + p^{13} T^{2} \) |
| 89 | \( 1 - 3502949738490 T + p^{13} T^{2} \) |
| 97 | \( 1 + 388932598558 T + p^{13} T^{2} \) |
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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
−26.13983667941665930077769816451, −24.31157294729847569717113019058, −22.83588606823754694134349459020, −20.50395468379408515166453894947, −19.35456790663496062114080756805, −15.80201722598698050092693916729, −14.28308720602488124131177581877, −11.81832745730909051908789154433, −7.992019600981742494142664392961, −3.64701944605928428755770205509,
3.64701944605928428755770205509, 7.992019600981742494142664392961, 11.81832745730909051908789154433, 14.28308720602488124131177581877, 15.80201722598698050092693916729, 19.35456790663496062114080756805, 20.50395468379408515166453894947, 22.83588606823754694134349459020, 24.31157294729847569717113019058, 26.13983667941665930077769816451
