| L(s) = 1 | + 32·2-s + 1.02e3·4-s − 1.08e3·5-s + 1.68e4·7-s + 3.27e4·8-s − 3.45e4·10-s − 4.54e5·11-s + 3.33e5·13-s + 5.37e5·14-s + 1.04e6·16-s − 3.29e5·17-s − 4.87e6·19-s − 1.10e6·20-s − 1.45e7·22-s + 4.05e6·23-s − 4.76e7·25-s + 1.06e7·26-s + 1.72e7·28-s − 6.48e7·29-s + 6.42e7·31-s + 3.35e7·32-s − 1.05e7·34-s − 1.81e7·35-s + 4.39e8·37-s − 1.56e8·38-s − 3.53e7·40-s − 1.14e9·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s − 0.154·5-s + 0.377·7-s + 0.353·8-s − 0.109·10-s − 0.851·11-s + 0.249·13-s + 0.267·14-s + 1/4·16-s − 0.0562·17-s − 0.451·19-s − 0.0772·20-s − 0.602·22-s + 0.131·23-s − 0.976·25-s + 0.176·26-s + 0.188·28-s − 0.586·29-s + 0.403·31-s + 0.176·32-s − 0.0397·34-s − 0.0584·35-s + 1.04·37-s − 0.319·38-s − 0.0546·40-s − 1.54·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(12-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+11/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(6)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{13}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{5} T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p^{5} T \) |
| good | 5 | \( 1 + 216 p T + p^{11} T^{2} \) |
| 11 | \( 1 + 454950 T + p^{11} T^{2} \) |
| 13 | \( 1 - 333386 T + p^{11} T^{2} \) |
| 17 | \( 1 + 329448 T + p^{11} T^{2} \) |
| 19 | \( 1 + 4876828 T + p^{11} T^{2} \) |
| 23 | \( 1 - 4054050 T + p^{11} T^{2} \) |
| 29 | \( 1 + 64835934 T + p^{11} T^{2} \) |
| 31 | \( 1 - 64240136 T + p^{11} T^{2} \) |
| 37 | \( 1 - 439691690 T + p^{11} T^{2} \) |
| 41 | \( 1 + 1147482084 T + p^{11} T^{2} \) |
| 43 | \( 1 - 682653668 T + p^{11} T^{2} \) |
| 47 | \( 1 + 2242598340 T + p^{11} T^{2} \) |
| 53 | \( 1 + 1692909282 T + p^{11} T^{2} \) |
| 59 | \( 1 - 856900848 T + p^{11} T^{2} \) |
| 61 | \( 1 - 328162910 T + p^{11} T^{2} \) |
| 67 | \( 1 + 6834378088 T + p^{11} T^{2} \) |
| 71 | \( 1 + 18096946722 T + p^{11} T^{2} \) |
| 73 | \( 1 - 4621386710 T + p^{11} T^{2} \) |
| 79 | \( 1 + 24901224268 T + p^{11} T^{2} \) |
| 83 | \( 1 + 24895768956 T + p^{11} T^{2} \) |
| 89 | \( 1 + 25356289956 T + p^{11} T^{2} \) |
| 97 | \( 1 + 40168146010 T + p^{11} T^{2} \) |
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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
−10.95449807904511349141825208442, −9.886985909977098957630365271683, −8.402766302201432146677019641779, −7.48912556936743564455122446952, −6.20748348631114732715149609030, −5.14242482605670659785271123740, −4.08334739941343455496079927494, −2.82145554856862899307997048311, −1.61815977544070517507414348368, 0,
1.61815977544070517507414348368, 2.82145554856862899307997048311, 4.08334739941343455496079927494, 5.14242482605670659785271123740, 6.20748348631114732715149609030, 7.48912556936743564455122446952, 8.402766302201432146677019641779, 9.886985909977098957630365271683, 10.95449807904511349141825208442
