| L(s) = 1 | − 12·2-s − 729·3-s − 8.04e3·4-s − 3.02e4·5-s + 8.74e3·6-s + 2.35e5·7-s + 1.94e5·8-s + 5.31e5·9-s + 3.62e5·10-s − 1.11e7·11-s + 5.86e6·12-s + 8.04e6·13-s − 2.82e6·14-s + 2.20e7·15-s + 6.35e7·16-s − 1.17e8·17-s − 6.37e6·18-s − 2.14e8·19-s + 2.43e8·20-s − 1.71e8·21-s + 1.34e8·22-s + 8.30e8·23-s − 1.42e8·24-s − 3.08e8·25-s − 9.65e7·26-s − 3.87e8·27-s − 1.89e9·28-s + ⋯ |
| L(s) = 1 | − 0.132·2-s − 0.577·3-s − 0.982·4-s − 0.864·5-s + 0.0765·6-s + 0.755·7-s + 0.262·8-s + 1/3·9-s + 0.114·10-s − 1.90·11-s + 0.567·12-s + 0.462·13-s − 0.100·14-s + 0.499·15-s + 0.947·16-s − 1.18·17-s − 0.0441·18-s − 1.04·19-s + 0.849·20-s − 0.436·21-s + 0.252·22-s + 1.16·23-s − 0.151·24-s − 0.252·25-s − 0.0613·26-s − 0.192·27-s − 0.741·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + p^{6} T \) |
| good | 2 | \( 1 + 3 p^{2} T + p^{13} T^{2} \) |
| 5 | \( 1 + 6042 p T + p^{13} T^{2} \) |
| 7 | \( 1 - 33584 p T + p^{13} T^{2} \) |
| 11 | \( 1 + 1016628 p T + p^{13} T^{2} \) |
| 13 | \( 1 - 8049614 T + p^{13} T^{2} \) |
| 17 | \( 1 + 117494622 T + p^{13} T^{2} \) |
| 19 | \( 1 + 214061380 T + p^{13} T^{2} \) |
| 23 | \( 1 - 830555544 T + p^{13} T^{2} \) |
| 29 | \( 1 + 1252400250 T + p^{13} T^{2} \) |
| 31 | \( 1 - 6159350552 T + p^{13} T^{2} \) |
| 37 | \( 1 + 5498191402 T + p^{13} T^{2} \) |
| 41 | \( 1 + 4678687878 T + p^{13} T^{2} \) |
| 43 | \( 1 - 7115013764 T + p^{13} T^{2} \) |
| 47 | \( 1 + 29528776992 T + p^{13} T^{2} \) |
| 53 | \( 1 + 204125042466 T + p^{13} T^{2} \) |
| 59 | \( 1 + 29909821020 T + p^{13} T^{2} \) |
| 61 | \( 1 + 134392006738 T + p^{13} T^{2} \) |
| 67 | \( 1 - 348518801948 T + p^{13} T^{2} \) |
| 71 | \( 1 - 1314335409192 T + p^{13} T^{2} \) |
| 73 | \( 1 + 1178875922326 T + p^{13} T^{2} \) |
| 79 | \( 1 + 1072420659640 T + p^{13} T^{2} \) |
| 83 | \( 1 - 1124025139644 T + p^{13} T^{2} \) |
| 89 | \( 1 - 2235610909530 T + p^{13} T^{2} \) |
| 97 | \( 1 + 14215257165502 T + p^{13} T^{2} \) |
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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
−23.01567458281830116463145667317, −21.16115388397722822760012940282, −18.85933439384294199835034224462, −17.58448488506871954593190645766, −15.47860897077394322959997490585, −13.09624541887251307495346647296, −10.88377145408859641382929243790, −8.169745648746111420252423993825, −4.74996296912952734958347812567, 0,
4.74996296912952734958347812567, 8.169745648746111420252423993825, 10.88377145408859641382929243790, 13.09624541887251307495346647296, 15.47860897077394322959997490585, 17.58448488506871954593190645766, 18.85933439384294199835034224462, 21.16115388397722822760012940282, 23.01567458281830116463145667317
