| L(s) = 1 | + (2.35 + 3.23i)2-s + (−6.79 − 2.20i)3-s + (−4.94 + 15.2i)4-s + (6.92 − 55.4i)5-s + (−8.82 − 27.1i)6-s − 47.7i·7-s + (−60.8 + 19.7i)8-s + (−155. − 112. i)9-s + (195. − 108. i)10-s + (400. − 290. i)11-s + (67.1 − 92.4i)12-s + (387. − 533. i)13-s + (154. − 112. i)14-s + (−169. + 361. i)15-s + (−207. − 150. i)16-s + (−120. + 39.2i)17-s + ⋯ |
| L(s) = 1 | + (0.415 + 0.572i)2-s + (−0.435 − 0.141i)3-s + (−0.154 + 0.475i)4-s + (0.123 − 0.992i)5-s + (−0.100 − 0.308i)6-s − 0.368i·7-s + (−0.336 + 0.109i)8-s + (−0.639 − 0.464i)9-s + (0.619 − 0.341i)10-s + (0.997 − 0.724i)11-s + (0.134 − 0.185i)12-s + (0.636 − 0.875i)13-s + (0.210 − 0.152i)14-s + (−0.194 + 0.414i)15-s + (−0.202 − 0.146i)16-s + (−0.101 + 0.0329i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.484 + 0.874i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.29578 - 0.763707i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.29578 - 0.763707i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.35 - 3.23i)T \) |
| 5 | \( 1 + (-6.92 + 55.4i)T \) |
| good | 3 | \( 1 + (6.79 + 2.20i)T + (196. + 142. i)T^{2} \) |
| 7 | \( 1 + 47.7iT - 1.68e4T^{2} \) |
| 11 | \( 1 + (-400. + 290. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (-387. + 533. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (120. - 39.2i)T + (1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (86.3 + 265. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (593. + 816. i)T + (-1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (1.43e3 - 4.42e3i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (2.70e3 + 8.32e3i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (5.52e3 - 7.60e3i)T + (-2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (1.24e3 + 902. i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 9.87e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-1.12e4 - 3.66e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-1.52e4 - 4.93e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (2.40e4 + 1.74e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-3.21e4 + 2.33e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-4.27e4 + 1.38e4i)T + (1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-2.04e4 + 6.28e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-4.11e4 - 5.66e4i)T + (-6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (5.46e3 - 1.68e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-3.49e4 + 1.13e4i)T + (3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (8.46e4 - 6.14e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (-9.71e4 - 3.15e4i)T + (6.94e9 + 5.04e9i)T^{2} \) |
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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
−14.27739758587623869806245114316, −13.25081870618721928748205664412, −12.22432330443711293781239479699, −11.10925725684569392027930497678, −9.202419786746597104413190348125, −8.194187416072592745140346494435, −6.42142081764122471894160127375, −5.41574900994884339141106971349, −3.74794982918304671271958345155, −0.73252129192551511933430585547,
2.10027328586492717037463568444, 3.89452736587214474523328905894, 5.66301638590987173178224062808, 6.89724754554031928787603325291, 8.965346428629563661166651611977, 10.32271162457010263837280114971, 11.35193800519088740695072957393, 12.09979255439146098687080071782, 13.78515083892247703294678664698, 14.47081768769145798173208920567
