| L(s) = 1 | + 128·2-s − 2.18e3·3-s + 1.63e4·4-s − 1.14e5·5-s − 2.79e5·6-s − 3.03e6·7-s + 2.09e6·8-s + 4.78e6·9-s − 1.46e7·10-s − 1.03e8·11-s − 3.58e7·12-s − 1.04e8·13-s − 3.88e8·14-s + 2.51e8·15-s + 2.68e8·16-s + 9.97e8·17-s + 6.12e8·18-s + 4.93e9·19-s − 1.88e9·20-s + 6.63e9·21-s − 1.32e10·22-s + 8.32e9·23-s − 4.58e9·24-s − 1.73e10·25-s − 1.33e10·26-s − 1.04e10·27-s − 4.97e10·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.657·5-s − 0.408·6-s − 1.39·7-s + 0.353·8-s + 1/3·9-s − 0.464·10-s − 1.60·11-s − 0.288·12-s − 0.461·13-s − 0.984·14-s + 0.379·15-s + 1/4·16-s + 0.589·17-s + 0.235·18-s + 1.26·19-s − 0.328·20-s + 0.804·21-s − 1.13·22-s + 0.509·23-s − 0.204·24-s − 0.568·25-s − 0.326·26-s − 0.192·27-s − 0.696·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(8)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{17}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{7} T \) |
| 3 | \( 1 + p^{7} T \) |
| good | 5 | \( 1 + 22962 p T + p^{15} T^{2} \) |
| 7 | \( 1 + 433504 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 9404700 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 104365834 T + p^{15} T^{2} \) |
| 17 | \( 1 - 997689762 T + p^{15} T^{2} \) |
| 19 | \( 1 - 4934015444 T + p^{15} T^{2} \) |
| 23 | \( 1 - 8324920200 T + p^{15} T^{2} \) |
| 29 | \( 1 - 104128242846 T + p^{15} T^{2} \) |
| 31 | \( 1 + 296696681512 T + p^{15} T^{2} \) |
| 37 | \( 1 + 178337455666 T + p^{15} T^{2} \) |
| 41 | \( 1 + 1790882416086 T + p^{15} T^{2} \) |
| 43 | \( 1 + 2863459422772 T + p^{15} T^{2} \) |
| 47 | \( 1 - 4332907521600 T + p^{15} T^{2} \) |
| 53 | \( 1 - 9732317104422 T + p^{15} T^{2} \) |
| 59 | \( 1 + 13514837176500 T + p^{15} T^{2} \) |
| 61 | \( 1 - 5352663511190 T + p^{15} T^{2} \) |
| 67 | \( 1 + 53233909720108 T + p^{15} T^{2} \) |
| 71 | \( 1 + 20229661643400 T + p^{15} T^{2} \) |
| 73 | \( 1 - 26264166466106 T + p^{15} T^{2} \) |
| 79 | \( 1 + 339031361615128 T + p^{15} T^{2} \) |
| 83 | \( 1 - 131684771045076 T + p^{15} T^{2} \) |
| 89 | \( 1 + 39352148322678 T + p^{15} T^{2} \) |
| 97 | \( 1 - 1128750908801474 T + p^{15} T^{2} \) |
| 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
−18.59638828057091361754563196311, −16.40622369628198614724642650675, −15.50932234651990188693513339586, −13.27043991391127981254467363707, −12.04994408283814145233740656932, −10.21600059651067033387315001498, −7.28892296803187118639224467436, −5.37856479530743569026031590586, −3.21748929729541942268506001835, 0,
3.21748929729541942268506001835, 5.37856479530743569026031590586, 7.28892296803187118639224467436, 10.21600059651067033387315001498, 12.04994408283814145233740656932, 13.27043991391127981254467363707, 15.50932234651990188693513339586, 16.40622369628198614724642650675, 18.59638828057091361754563196311
