| L(s) = 1 | − 128·2-s − 6.25e3·3-s + 1.63e4·4-s − 9.05e4·5-s + 8.00e5·6-s − 2.09e6·8-s + 2.47e7·9-s + 1.15e7·10-s − 9.58e7·11-s − 1.02e8·12-s + 5.97e7·13-s + 5.65e8·15-s + 2.68e8·16-s + 1.35e9·17-s − 3.16e9·18-s − 3.78e9·19-s − 1.48e9·20-s + 1.22e10·22-s − 1.16e10·23-s + 1.31e10·24-s − 2.23e10·25-s − 7.65e9·26-s − 6.49e10·27-s − 2.89e10·29-s − 7.24e10·30-s − 2.53e11·31-s − 3.43e10·32-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.65·3-s + 1/2·4-s − 0.518·5-s + 1.16·6-s − 0.353·8-s + 1.72·9-s + 0.366·10-s − 1.48·11-s − 0.825·12-s + 0.264·13-s + 0.855·15-s + 1/4·16-s + 0.801·17-s − 1.21·18-s − 0.971·19-s − 0.259·20-s + 1.04·22-s − 0.710·23-s + 0.583·24-s − 0.731·25-s − 0.186·26-s − 1.19·27-s − 0.311·29-s − 0.604·30-s − 1.65·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(16-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 98 ^{s/2} \, \Gamma_{\C}(s+15/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + p^{7} T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + 2084 p T + p^{15} T^{2} \) |
| 5 | \( 1 + 18102 p T + p^{15} T^{2} \) |
| 11 | \( 1 + 8717268 p T + p^{15} T^{2} \) |
| 13 | \( 1 - 4598626 p T + p^{15} T^{2} \) |
| 17 | \( 1 - 1355814414 T + p^{15} T^{2} \) |
| 19 | \( 1 + 3783593180 T + p^{15} T^{2} \) |
| 23 | \( 1 + 11608845528 T + p^{15} T^{2} \) |
| 29 | \( 1 + 28959105930 T + p^{15} T^{2} \) |
| 31 | \( 1 + 253685353952 T + p^{15} T^{2} \) |
| 37 | \( 1 - 817641294446 T + p^{15} T^{2} \) |
| 41 | \( 1 - 682333284198 T + p^{15} T^{2} \) |
| 43 | \( 1 - 366945604292 T + p^{15} T^{2} \) |
| 47 | \( 1 + 695741581776 T + p^{15} T^{2} \) |
| 53 | \( 1 - 12993372468702 T + p^{15} T^{2} \) |
| 59 | \( 1 + 9209035340340 T + p^{15} T^{2} \) |
| 61 | \( 1 - 42338641200298 T + p^{15} T^{2} \) |
| 67 | \( 1 - 448205790308 p T + p^{15} T^{2} \) |
| 71 | \( 1 - 115328696975352 T + p^{15} T^{2} \) |
| 73 | \( 1 + 43787346432122 T + p^{15} T^{2} \) |
| 79 | \( 1 - 79603813043120 T + p^{15} T^{2} \) |
| 83 | \( 1 - 41169504396 p T + p^{15} T^{2} \) |
| 89 | \( 1 - 377306179184790 T + p^{15} T^{2} \) |
| 97 | \( 1 - 166982186657374 T + p^{15} T^{2} \) |
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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
−10.68664241973426545887009073319, −9.784680247756129141382463456900, −8.140473901953274364287365494842, −7.30764864329566212130167705235, −6.04036046777216732881960280913, −5.30523213589487418116967815539, −3.93236253625903802660221559132, −2.17020575646353006229707871043, −0.73557680670821467177626155422, 0,
0.73557680670821467177626155422, 2.17020575646353006229707871043, 3.93236253625903802660221559132, 5.30523213589487418116967815539, 6.04036046777216732881960280913, 7.30764864329566212130167705235, 8.140473901953274364287365494842, 9.784680247756129141382463456900, 10.68664241973426545887009073319
