| L(s) = 1 | + (5.22 − 1.61i)2-s + (8.72 + 22.2i)3-s + (−1.77 + 1.20i)4-s + (−104. − 15.7i)5-s + (81.3 + 102. i)6-s + (122. + 42.4i)7-s + (−116. + 145. i)8-s + (−240. + 223. i)9-s + (−572. + 86.2i)10-s + (99.8 + 92.6i)11-s + (−42.3 − 28.8i)12-s + (62.1 − 272. i)13-s + (707. + 24.4i)14-s + (−563. − 2.46e3i)15-s + (−347. + 885. i)16-s + (−58.6 + 783. i)17-s + ⋯ |
| L(s) = 1 | + (0.923 − 0.284i)2-s + (0.559 + 1.42i)3-s + (−0.0553 + 0.0377i)4-s + (−1.87 − 0.282i)5-s + (0.923 + 1.15i)6-s + (0.944 + 0.327i)7-s + (−0.642 + 0.805i)8-s + (−0.989 + 0.917i)9-s + (−1.80 + 0.272i)10-s + (0.248 + 0.230i)11-s + (−0.0848 − 0.0578i)12-s + (0.101 − 0.446i)13-s + (0.965 + 0.0333i)14-s + (−0.646 − 2.83i)15-s + (−0.339 + 0.864i)16-s + (−0.0492 + 0.657i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.943854 + 1.70838i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.943854 + 1.70838i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-122. - 42.4i)T \) |
| good | 2 | \( 1 + (-5.22 + 1.61i)T + (26.4 - 18.0i)T^{2} \) |
| 3 | \( 1 + (-8.72 - 22.2i)T + (-178. + 165. i)T^{2} \) |
| 5 | \( 1 + (104. + 15.7i)T + (2.98e3 + 921. i)T^{2} \) |
| 11 | \( 1 + (-99.8 - 92.6i)T + (1.20e4 + 1.60e5i)T^{2} \) |
| 13 | \( 1 + (-62.1 + 272. i)T + (-3.34e5 - 1.61e5i)T^{2} \) |
| 17 | \( 1 + (58.6 - 783. i)T + (-1.40e6 - 2.11e5i)T^{2} \) |
| 19 | \( 1 + (-1.38e3 - 2.39e3i)T + (-1.23e6 + 2.14e6i)T^{2} \) |
| 23 | \( 1 + (236. + 3.15e3i)T + (-6.36e6 + 9.59e5i)T^{2} \) |
| 29 | \( 1 + (-4.71e3 + 2.26e3i)T + (1.27e7 - 1.60e7i)T^{2} \) |
| 31 | \( 1 + (2.55e3 - 4.42e3i)T + (-1.43e7 - 2.47e7i)T^{2} \) |
| 37 | \( 1 + (1.88e3 + 1.28e3i)T + (2.53e7 + 6.45e7i)T^{2} \) |
| 41 | \( 1 + (5.79e3 - 7.26e3i)T + (-2.57e7 - 1.12e8i)T^{2} \) |
| 43 | \( 1 + (8.40e3 + 1.05e4i)T + (-3.27e7 + 1.43e8i)T^{2} \) |
| 47 | \( 1 + (4.89e3 - 1.51e3i)T + (1.89e8 - 1.29e8i)T^{2} \) |
| 53 | \( 1 + (-1.21e4 + 8.26e3i)T + (1.52e8 - 3.89e8i)T^{2} \) |
| 59 | \( 1 + (7.34e3 - 1.10e3i)T + (6.83e8 - 2.10e8i)T^{2} \) |
| 61 | \( 1 + (-9.82e3 - 6.69e3i)T + (3.08e8 + 7.86e8i)T^{2} \) |
| 67 | \( 1 + (-4.96e3 + 8.59e3i)T + (-6.75e8 - 1.16e9i)T^{2} \) |
| 71 | \( 1 + (2.05e4 + 9.90e3i)T + (1.12e9 + 1.41e9i)T^{2} \) |
| 73 | \( 1 + (1.38e4 + 4.27e3i)T + (1.71e9 + 1.16e9i)T^{2} \) |
| 79 | \( 1 + (-4.31e4 - 7.46e4i)T + (-1.53e9 + 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-492. - 2.15e3i)T + (-3.54e9 + 1.70e9i)T^{2} \) |
| 89 | \( 1 + (2.30e3 - 2.13e3i)T + (4.17e8 - 5.56e9i)T^{2} \) |
| 97 | \( 1 - 1.65e5T + 8.58e9T^{2} \) |
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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
−14.88314629322674606889324230956, −14.31574711838116275673161679754, −12.40974700404112116222600064979, −11.71608139763792213682908538468, −10.50627400217049890525185787684, −8.625163143129392673570587849666, −8.116088210581519164965973873330, −5.05004382511845517783943195156, −4.18491850096877613688237322586, −3.35177886952115890307566984523,
0.76154364176572218753187666071, 3.29962693502789694151321434494, 4.73761932669662779602835475335, 6.91581558634555231880520277382, 7.56053663229898361942920141714, 8.788928585685443676952911916437, 11.44657230668083340112240698585, 11.96057904813145789023340408521, 13.33264035578868908634176586580, 14.08264162140998563000408282294
