| L(s) = 1 | + 10·2-s + 27·3-s − 28·4-s − 410·5-s + 270·6-s − 1.02e3·7-s − 1.56e3·8-s + 729·9-s − 4.10e3·10-s − 1.33e3·11-s − 756·12-s + 1.29e4·13-s − 1.02e4·14-s − 1.10e4·15-s − 1.20e4·16-s + 1.70e4·17-s + 7.29e3·18-s − 5.41e4·19-s + 1.14e4·20-s − 2.77e4·21-s − 1.33e4·22-s − 1.14e4·23-s − 4.21e4·24-s + 8.99e4·25-s + 1.29e5·26-s + 1.96e4·27-s + 2.87e4·28-s + ⋯ |
| L(s) = 1 | + 0.883·2-s + 0.577·3-s − 0.218·4-s − 1.46·5-s + 0.510·6-s − 1.13·7-s − 1.07·8-s + 1/3·9-s − 1.29·10-s − 0.301·11-s − 0.126·12-s + 1.63·13-s − 1.00·14-s − 0.846·15-s − 0.733·16-s + 0.842·17-s + 0.294·18-s − 1.81·19-s + 0.320·20-s − 0.654·21-s − 0.266·22-s − 0.196·23-s − 0.621·24-s + 1.15·25-s + 1.44·26-s + 0.192·27-s + 0.247·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - p^{3} T \) |
| 11 | \( 1 + p^{3} T \) |
| good | 2 | \( 1 - 5 p T + p^{7} T^{2} \) |
| 5 | \( 1 + 82 p T + p^{7} T^{2} \) |
| 7 | \( 1 + 1028 T + p^{7} T^{2} \) |
| 13 | \( 1 - 12958 T + p^{7} T^{2} \) |
| 17 | \( 1 - 17062 T + p^{7} T^{2} \) |
| 19 | \( 1 + 54168 T + p^{7} T^{2} \) |
| 23 | \( 1 + 11488 T + p^{7} T^{2} \) |
| 29 | \( 1 + 186654 T + p^{7} T^{2} \) |
| 31 | \( 1 + 188672 T + p^{7} T^{2} \) |
| 37 | \( 1 - 395886 T + p^{7} T^{2} \) |
| 41 | \( 1 + 47546 T + p^{7} T^{2} \) |
| 43 | \( 1 - 602088 T + p^{7} T^{2} \) |
| 47 | \( 1 + 647200 T + p^{7} T^{2} \) |
| 53 | \( 1 + 1312722 T + p^{7} T^{2} \) |
| 59 | \( 1 + 2681140 T + p^{7} T^{2} \) |
| 61 | \( 1 - 551190 T + p^{7} T^{2} \) |
| 67 | \( 1 - 459260 T + p^{7} T^{2} \) |
| 71 | \( 1 + 18072 T + p^{7} T^{2} \) |
| 73 | \( 1 + 426062 T + p^{7} T^{2} \) |
| 79 | \( 1 - 297764 T + p^{7} T^{2} \) |
| 83 | \( 1 - 5684028 T + p^{7} T^{2} \) |
| 89 | \( 1 + 6342966 T + p^{7} T^{2} \) |
| 97 | \( 1 - 16651586 T + p^{7} T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.62297400172239789414015005037, −13.15234304379254084387411735575, −12.59609630062394849309637652179, −11.04406848076407803940761174014, −9.175493062834662677676107439120, −7.976772153457015728592745423836, −6.18294940029144920899708516586, −4.07109170505974689466733458674, −3.35700454486395836232459126376, 0,
3.35700454486395836232459126376, 4.07109170505974689466733458674, 6.18294940029144920899708516586, 7.976772153457015728592745423836, 9.175493062834662677676107439120, 11.04406848076407803940761174014, 12.59609630062394849309637652179, 13.15234304379254084387411735575, 14.62297400172239789414015005037
