| L(s) = 1 | + 64·2-s + 1.02e3·3-s + 4.09e3·4-s − 4.32e3·5-s + 6.56e4·6-s + 2.62e5·8-s − 5.41e5·9-s − 2.76e5·10-s − 8.78e6·11-s + 4.20e6·12-s + 2.04e7·13-s − 4.43e6·15-s + 1.67e7·16-s − 1.71e6·17-s − 3.46e7·18-s + 1.09e8·19-s − 1.76e7·20-s − 5.62e8·22-s − 6.46e8·23-s + 2.68e8·24-s − 1.20e9·25-s + 1.30e9·26-s − 2.19e9·27-s + 7.28e8·29-s − 2.83e8·30-s − 1.02e9·31-s + 1.07e9·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.812·3-s + 1/2·4-s − 0.123·5-s + 0.574·6-s + 0.353·8-s − 0.339·9-s − 0.0874·10-s − 1.49·11-s + 0.406·12-s + 1.17·13-s − 0.100·15-s + 1/4·16-s − 0.0172·17-s − 0.240·18-s + 0.534·19-s − 0.0618·20-s − 1.05·22-s − 0.910·23-s + 0.287·24-s − 0.984·25-s + 0.829·26-s − 1.08·27-s + 0.227·29-s − 0.0710·30-s − 0.208·31-s + 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 - p^{6} T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 38 p^{3} T + p^{13} T^{2} \) |
| 5 | \( 1 + 864 p T + p^{13} T^{2} \) |
| 11 | \( 1 + 8787312 T + p^{13} T^{2} \) |
| 13 | \( 1 - 20420932 T + p^{13} T^{2} \) |
| 17 | \( 1 + 1719462 T + p^{13} T^{2} \) |
| 19 | \( 1 - 109702942 T + p^{13} T^{2} \) |
| 23 | \( 1 + 646760160 T + p^{13} T^{2} \) |
| 29 | \( 1 - 728867274 T + p^{13} T^{2} \) |
| 31 | \( 1 + 1028049116 T + p^{13} T^{2} \) |
| 37 | \( 1 - 14229390962 T + p^{13} T^{2} \) |
| 41 | \( 1 + 44544458406 T + p^{13} T^{2} \) |
| 43 | \( 1 + 54689828968 T + p^{13} T^{2} \) |
| 47 | \( 1 + 47868325716 T + p^{13} T^{2} \) |
| 53 | \( 1 + 169986882858 T + p^{13} T^{2} \) |
| 59 | \( 1 - 300765540198 T + p^{13} T^{2} \) |
| 61 | \( 1 + 369996272360 T + p^{13} T^{2} \) |
| 67 | \( 1 + 787010801908 T + p^{13} T^{2} \) |
| 71 | \( 1 - 559441472256 T + p^{13} T^{2} \) |
| 73 | \( 1 + 121137579650 T + p^{13} T^{2} \) |
| 79 | \( 1 - 290426785064 T + p^{13} T^{2} \) |
| 83 | \( 1 - 3965105603046 T + p^{13} T^{2} \) |
| 89 | \( 1 - 6025919250630 T + p^{13} T^{2} \) |
| 97 | \( 1 + 11302818199190 T + p^{13} T^{2} \) |
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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
−10.96191086815311281392396586366, −9.814726943352536811438448369357, −8.365146421227729563338690231930, −7.76140655401260873924454210859, −6.19757622734137314531669608910, −5.16525417537185988427005367429, −3.73271819760608123756051271682, −2.89233569789010415860320474415, −1.77629634777287443461320412202, 0,
1.77629634777287443461320412202, 2.89233569789010415860320474415, 3.73271819760608123756051271682, 5.16525417537185988427005367429, 6.19757622734137314531669608910, 7.76140655401260873924454210859, 8.365146421227729563338690231930, 9.814726943352536811438448369357, 10.96191086815311281392396586366
