L(s) = 1 | + 64·2-s − 1.02e3·3-s + 4.09e3·4-s + 4.32e3·5-s − 6.56e4·6-s + 1.17e5·7-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 + 7.52e6·14-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 − 1.20e8·21-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 + 4.81e8·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 0.812·3-s + 1/2·4-s + 0.123·5-s − 0.574·6-s + 0.377·7-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 + 0.267·14-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 − 0.307·21-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 + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p^{6} T \) |
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
−15.63705033608913310438039885834, −14.24971356415454753954880496937, −12.75807747849829358787186789012, −11.54949828374638720530535194494, −10.26398123799017957117895697677, −7.79879227796490286493012193853, −5.92500521625767098203232889554, −4.77489662943688934543950811295, −2.45459606414312679081256500314, 0,
2.45459606414312679081256500314, 4.77489662943688934543950811295, 5.92500521625767098203232889554, 7.79879227796490286493012193853, 10.26398123799017957117895697677, 11.54949828374638720530535194494, 12.75807747849829358787186789012, 14.24971356415454753954880496937, 15.63705033608913310438039885834