| L(s) = 1 | + 234·2-s + 2.18e3·3-s + 2.19e4·4-s + 5.11e5·6-s + 1.37e6·7-s − 2.52e6·8-s + 4.78e6·9-s + 3.40e7·11-s + 4.80e7·12-s − 3.84e8·13-s + 3.21e8·14-s − 1.31e9·16-s − 1.25e9·17-s + 1.11e9·18-s − 2.49e9·19-s + 3.00e9·21-s + 7.96e9·22-s − 1.12e10·23-s − 5.51e9·24-s − 8.98e10·26-s + 1.04e10·27-s + 3.01e10·28-s − 4.84e10·29-s + 1.30e11·31-s − 2.24e11·32-s + 7.44e10·33-s − 2.94e11·34-s + ⋯ |
| L(s) = 1 | + 1.29·2-s + 0.577·3-s + 0.671·4-s + 0.746·6-s + 0.630·7-s − 0.425·8-s + 1/3·9-s + 0.526·11-s + 0.387·12-s − 1.69·13-s + 0.814·14-s − 1.22·16-s − 0.744·17-s + 0.430·18-s − 0.641·19-s + 0.363·21-s + 0.680·22-s − 0.691·23-s − 0.245·24-s − 2.19·26-s + 0.192·27-s + 0.422·28-s − 0.521·29-s + 0.852·31-s − 1.15·32-s + 0.303·33-s − 0.962·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - p^{7} T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 117 p T + p^{15} T^{2} \) |
| 7 | \( 1 - 196192 p T + p^{15} T^{2} \) |
| 11 | \( 1 - 3093732 p T + p^{15} T^{2} \) |
| 13 | \( 1 + 29540174 p T + p^{15} T^{2} \) |
| 17 | \( 1 + 1259207586 T + p^{15} T^{2} \) |
| 19 | \( 1 + 2499071020 T + p^{15} T^{2} \) |
| 23 | \( 1 + 11284833672 T + p^{15} T^{2} \) |
| 29 | \( 1 + 48413458530 T + p^{15} T^{2} \) |
| 31 | \( 1 - 130547265752 T + p^{15} T^{2} \) |
| 37 | \( 1 - 200223317554 T + p^{15} T^{2} \) |
| 41 | \( 1 - 679141724202 T + p^{15} T^{2} \) |
| 43 | \( 1 + 279482194892 T + p^{15} T^{2} \) |
| 47 | \( 1 + 1520672832576 T + p^{15} T^{2} \) |
| 53 | \( 1 + 2646053822502 T + p^{15} T^{2} \) |
| 59 | \( 1 - 7399371294540 T + p^{15} T^{2} \) |
| 61 | \( 1 + 42659617819498 T + p^{15} T^{2} \) |
| 67 | \( 1 - 56408026065964 T + p^{15} T^{2} \) |
| 71 | \( 1 + 133149677299848 T + p^{15} T^{2} \) |
| 73 | \( 1 + 105603350884922 T + p^{15} T^{2} \) |
| 79 | \( 1 + 55665674361880 T + p^{15} T^{2} \) |
| 83 | \( 1 + 378077412997332 T + p^{15} T^{2} \) |
| 89 | \( 1 - 219315065897610 T + p^{15} T^{2} \) |
| 97 | \( 1 + 703322682162626 T + p^{15} T^{2} \) |
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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
−11.40441023625340027501249049214, −9.863096849178173428171954311734, −8.734504513965415346346539933924, −7.43435782586256507292966816405, −6.21616492168856685287367476615, −4.82707543334138874237734185551, −4.20892663542519355777374993318, −2.84238810475826671085658024914, −1.90454974341686018409892017704, 0,
1.90454974341686018409892017704, 2.84238810475826671085658024914, 4.20892663542519355777374993318, 4.82707543334138874237734185551, 6.21616492168856685287367476615, 7.43435782586256507292966816405, 8.734504513965415346346539933924, 9.863096849178173428171954311734, 11.40441023625340027501249049214
