| L(s) = 1 | + (−7.56 + 2.33i)2-s + (−5.53 − 14.0i)3-s + (25.3 − 17.2i)4-s + (−100. − 15.0i)5-s + (74.6 + 93.6i)6-s + (−91.8 − 91.5i)7-s + (6.78 − 8.50i)8-s + (10.1 − 9.42i)9-s + (792. − 119. i)10-s + (−221. − 205. i)11-s + (−383. − 261. i)12-s + (−175. + 767. i)13-s + (907. + 477. i)14-s + (341. + 1.49e3i)15-s + (−389. + 992. i)16-s + (53.0 − 708. i)17-s + ⋯ |
| L(s) = 1 | + (−1.33 + 0.412i)2-s + (−0.354 − 0.903i)3-s + (0.790 − 0.539i)4-s + (−1.79 − 0.269i)5-s + (0.847 + 1.06i)6-s + (−0.708 − 0.705i)7-s + (0.0374 − 0.0469i)8-s + (0.0417 − 0.0387i)9-s + (2.50 − 0.377i)10-s + (−0.551 − 0.511i)11-s + (−0.767 − 0.523i)12-s + (−0.287 + 1.25i)13-s + (1.23 + 0.651i)14-s + (0.391 + 1.71i)15-s + (−0.380 + 0.969i)16-s + (0.0445 − 0.594i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.332 - 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0832246 + 0.0589368i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0832246 + 0.0589368i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (91.8 + 91.5i)T \) |
| good | 2 | \( 1 + (7.56 - 2.33i)T + (26.4 - 18.0i)T^{2} \) |
| 3 | \( 1 + (5.53 + 14.0i)T + (-178. + 165. i)T^{2} \) |
| 5 | \( 1 + (100. + 15.0i)T + (2.98e3 + 921. i)T^{2} \) |
| 11 | \( 1 + (221. + 205. i)T + (1.20e4 + 1.60e5i)T^{2} \) |
| 13 | \( 1 + (175. - 767. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + (-53.0 + 708. i)T + (-1.40e6 - 2.11e5i)T^{2} \) |
| 19 | \( 1 + (-713. - 1.23e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (89.7 + 1.19e3i)T + (-6.36e6 + 9.59e5i)T^{2} \) |
| 29 | \( 1 + (390. - 188. i)T + (1.27e7 - 1.60e7i)T^{2} \) |
| 31 | \( 1 + (-2.71e3 + 4.70e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (-3.78e3 - 2.57e3i)T + (2.53e7 + 6.45e7i)T^{2} \) |
| 41 | \( 1 + (3.80e3 - 4.76e3i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 43 | \( 1 + (1.44e4 + 1.80e4i)T + (-3.27e7 + 1.43e8i)T^{2} \) |
| 47 | \( 1 + (1.34e3 - 414. i)T + (1.89e8 - 1.29e8i)T^{2} \) |
| 53 | \( 1 + (-1.60e4 + 1.09e4i)T + (1.52e8 - 3.89e8i)T^{2} \) |
| 59 | \( 1 + (4.07e4 - 6.13e3i)T + (6.83e8 - 2.10e8i)T^{2} \) |
| 61 | \( 1 + (-4.27e4 - 2.91e4i)T + (3.08e8 + 7.86e8i)T^{2} \) |
| 67 | \( 1 + (3.60e3 - 6.23e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (-4.11e4 - 1.98e4i)T + (1.12e9 + 1.41e9i)T^{2} \) |
| 73 | \( 1 + (2.35e4 + 7.26e3i)T + (1.71e9 + 1.16e9i)T^{2} \) |
| 79 | \( 1 + (-4.50e3 - 7.80e3i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-1.66e4 - 7.27e4i)T + (-3.54e9 + 1.70e9i)T^{2} \) |
| 89 | \( 1 + (9.10e4 - 8.45e4i)T + (4.17e8 - 5.56e9i)T^{2} \) |
| 97 | \( 1 + 1.43e5T + 8.58e9T^{2} \) |
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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
−15.38298552200850708843432830339, −13.49614089005738357923807902678, −12.25323430315577349972674468222, −11.32736601074886222295309016119, −9.836549123060366569761734376216, −8.374792934772267911285191148186, −7.43086484140279763974776000723, −6.75418605285498780177703779354, −4.00260525556600415505576007664, −0.821445905606465110224681048382,
0.13048253420679961909931122913, 3.13506486456294483684658202746, 4.92336225986220204648946160522, 7.39397318743355995695376416266, 8.356175788541134684338626587011, 9.754269999162464450927674936086, 10.62863178403994901688901094258, 11.54367914471233166722837774156, 12.69571986792945988261629819904, 15.21019044683836106496200304923
