| L(s) = 1 | + (1.72 − 2.13i)2-s + (−0.838 − 0.544i)3-s + (−1.15 − 5.42i)4-s + (1.89 + 1.19i)5-s + (−2.61 + 0.848i)6-s + (2.64 − 0.172i)7-s + (−8.68 − 4.42i)8-s + (0.406 + 0.913i)9-s + (5.81 − 1.97i)10-s + (−4.35 − 1.93i)11-s + (−1.98 + 5.17i)12-s + (−0.362 − 2.29i)13-s + (4.19 − 5.93i)14-s + (−0.936 − 2.03i)15-s + (−14.3 + 6.37i)16-s + (−0.0671 − 1.28i)17-s + ⋯ |
| L(s) = 1 | + (1.22 − 1.50i)2-s + (−0.484 − 0.314i)3-s + (−0.576 − 2.71i)4-s + (0.845 + 0.533i)5-s + (−1.06 + 0.346i)6-s + (0.997 − 0.0651i)7-s + (−3.06 − 1.56i)8-s + (0.135 + 0.304i)9-s + (1.83 − 0.624i)10-s + (−1.31 − 0.584i)11-s + (−0.573 + 1.49i)12-s + (−0.100 − 0.635i)13-s + (1.12 − 1.58i)14-s + (−0.241 − 0.524i)15-s + (−3.57 + 1.59i)16-s + (−0.0162 − 0.310i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 + 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.308932 - 2.48268i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.308932 - 2.48268i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.838 + 0.544i)T \) |
| 5 | \( 1 + (-1.89 - 1.19i)T \) |
| 7 | \( 1 + (-2.64 + 0.172i)T \) |
| good | 2 | \( 1 + (-1.72 + 2.13i)T + (-0.415 - 1.95i)T^{2} \) |
| 11 | \( 1 + (4.35 + 1.93i)T + (7.36 + 8.17i)T^{2} \) |
| 13 | \( 1 + (0.362 + 2.29i)T + (-12.3 + 4.01i)T^{2} \) |
| 17 | \( 1 + (0.0671 + 1.28i)T + (-16.9 + 1.77i)T^{2} \) |
| 19 | \( 1 + (-1.64 - 0.348i)T + (17.3 + 7.72i)T^{2} \) |
| 23 | \( 1 + (-5.13 - 4.15i)T + (4.78 + 22.4i)T^{2} \) |
| 29 | \( 1 + (1.14 + 0.371i)T + (23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.28 - 4.75i)T + (3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (3.13 + 1.20i)T + (27.4 + 24.7i)T^{2} \) |
| 41 | \( 1 + (3.20 - 4.41i)T + (-12.6 - 38.9i)T^{2} \) |
| 43 | \( 1 + (-3.32 + 3.32i)T - 43iT^{2} \) |
| 47 | \( 1 + (-7.90 - 0.414i)T + (46.7 + 4.91i)T^{2} \) |
| 53 | \( 1 + (6.98 - 10.7i)T + (-21.5 - 48.4i)T^{2} \) |
| 59 | \( 1 + (-1.24 + 11.8i)T + (-57.7 - 12.2i)T^{2} \) |
| 61 | \( 1 + (3.05 - 0.321i)T + (59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (0.120 - 0.00631i)T + (66.6 - 7.00i)T^{2} \) |
| 71 | \( 1 + (-1.42 + 4.38i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.0586 - 0.152i)T + (-54.2 + 48.8i)T^{2} \) |
| 79 | \( 1 + (-8.79 + 7.91i)T + (8.25 - 78.5i)T^{2} \) |
| 83 | \( 1 + (-0.654 + 1.28i)T + (-48.7 - 67.1i)T^{2} \) |
| 89 | \( 1 + (-0.594 - 5.65i)T + (-87.0 + 18.5i)T^{2} \) |
| 97 | \( 1 + (-8.30 - 16.3i)T + (-57.0 + 78.4i)T^{2} \) |
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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
−10.73610602853960441900744350673, −10.22798833972309136668159450054, −9.106094315929280648405964545311, −7.59799572795365485330292895543, −6.23248420609245040406668135682, −5.26396011433341029659508961640, −5.01639519905654665698268026361, −3.27518246625515274389177675847, −2.41489888112512917100443945261, −1.17479081327488758527658174969,
2.50940648054370519185299806669, 4.28367616236826006454786834907, 4.97198328768959596236108526820, 5.45973476988491133351662185611, 6.47141209966147755741758870073, 7.42574915219889023115322520366, 8.319868290937926968840730619505, 9.145654796555298311425897255383, 10.42331172507089476209664228480, 11.60289646135523842321754326473
