| L(s) = 1 | + (0.0686 + 0.653i)2-s + (1.27 − 1.42i)3-s + (1.53 − 0.326i)4-s + (−3.26 + 1.45i)5-s + (1.01 + 0.738i)6-s + (−2.40 + 1.09i)7-s + (0.724 + 2.22i)8-s + (−0.0686 − 0.653i)9-s + (−1.17 − 2.02i)10-s + (1.5 − 2.59i)12-s + (−4.78 + 3.47i)13-s + (−0.879 − 1.49i)14-s + (−2.10 + 6.49i)15-s + (1.46 − 0.650i)16-s + (−0.173 + 1.64i)17-s + (0.421 − 0.0896i)18-s + ⋯ |
| L(s) = 1 | + (0.0485 + 0.461i)2-s + (0.738 − 0.820i)3-s + (0.767 − 0.163i)4-s + (−1.45 + 0.649i)5-s + (0.414 + 0.301i)6-s + (−0.910 + 0.413i)7-s + (0.256 + 0.787i)8-s + (−0.0228 − 0.217i)9-s + (−0.370 − 0.641i)10-s + (0.433 − 0.749i)12-s + (−1.32 + 0.963i)13-s + (−0.235 − 0.400i)14-s + (−0.544 + 1.67i)15-s + (0.365 − 0.162i)16-s + (−0.0419 + 0.399i)17-s + (0.0994 − 0.0211i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 847 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.508 - 0.861i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.575895 + 1.00887i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.575895 + 1.00887i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (2.40 - 1.09i)T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + (-0.0686 - 0.653i)T + (-1.95 + 0.415i)T^{2} \) |
| 3 | \( 1 + (-1.27 + 1.42i)T + (-0.313 - 2.98i)T^{2} \) |
| 5 | \( 1 + (3.26 - 1.45i)T + (3.34 - 3.71i)T^{2} \) |
| 13 | \( 1 + (4.78 - 3.47i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (0.173 - 1.64i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (-1.44 - 0.307i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (1.67 - 2.89i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.951 - 2.92i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (6.46 + 2.87i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (3.01 + 3.35i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (-0.397 - 1.22i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 1.59T + 43T^{2} \) |
| 47 | \( 1 + (-1.62 - 0.344i)T + (42.9 + 19.1i)T^{2} \) |
| 53 | \( 1 + (-8.42 - 3.75i)T + (35.4 + 39.3i)T^{2} \) |
| 59 | \( 1 + (8.65 - 1.84i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (-6.10 + 2.71i)T + (40.8 - 45.3i)T^{2} \) |
| 67 | \( 1 + (-4.91 - 8.50i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.97 - 5.06i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.46 + 0.948i)T + (66.6 - 29.6i)T^{2} \) |
| 79 | \( 1 + (0.668 + 6.35i)T + (-77.2 + 16.4i)T^{2} \) |
| 83 | \( 1 + (-0.135 - 0.0986i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-1.28 + 2.22i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.87 - 5.72i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.58255339268716080310989097633, −9.479032202944544629919703145819, −8.461612570579504380860254660106, −7.54136844287800043658381186189, −7.23160408339741433776962416759, −6.61615737183595309512046413411, −5.38304609074527439504589349067, −3.88461892320387088923488891897, −2.88830406562608099447707713457, −2.03329333367379665593825168273,
0.46783707199136722260994173490, 2.65874392137752348941418219676, 3.46826208861896158032913123955, 4.04610592416039114009319048737, 5.12888817937985222051103210770, 6.73045199092899701221766775109, 7.50139861756659341482385711434, 8.177230695422003914351474130601, 9.232489336161546241842248792200, 9.967484197928675358261718335338
