| L(s) = 1 | + (−3.23 − 2.35i)2-s + (−3.10 − 9.55i)3-s + (4.94 + 15.2i)4-s + (−0.387 − 55.9i)5-s + (−12.4 + 38.2i)6-s − 100.·7-s + (19.7 − 60.8i)8-s + (114. − 83.4i)9-s + (−130. + 181. i)10-s + (−8.34 − 6.06i)11-s + (130. − 94.5i)12-s + (−905. + 657. i)13-s + (326. + 236. i)14-s + (−533. + 177. i)15-s + (−207. + 150. i)16-s + (−455. + 1.40e3i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 − 0.415i)2-s + (−0.199 − 0.613i)3-s + (0.154 + 0.475i)4-s + (−0.00692 − 0.999i)5-s + (−0.140 + 0.433i)6-s − 0.777·7-s + (0.109 − 0.336i)8-s + (0.472 − 0.343i)9-s + (−0.411 + 0.574i)10-s + (−0.0207 − 0.0151i)11-s + (0.260 − 0.189i)12-s + (−1.48 + 1.07i)13-s + (0.444 + 0.323i)14-s + (−0.611 + 0.203i)15-s + (−0.202 + 0.146i)16-s + (−0.382 + 1.17i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.738 - 0.674i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.107027 + 0.275889i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.107027 + 0.275889i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (3.23 + 2.35i)T \) |
| 5 | \( 1 + (0.387 + 55.9i)T \) |
| good | 3 | \( 1 + (3.10 + 9.55i)T + (-196. + 142. i)T^{2} \) |
| 7 | \( 1 + 100.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (8.34 + 6.06i)T + (4.97e4 + 1.53e5i)T^{2} \) |
| 13 | \( 1 + (905. - 657. i)T + (1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (455. - 1.40e3i)T + (-1.14e6 - 8.34e5i)T^{2} \) |
| 19 | \( 1 + (71.1 - 219. i)T + (-2.00e6 - 1.45e6i)T^{2} \) |
| 23 | \( 1 + (466. + 339. i)T + (1.98e6 + 6.12e6i)T^{2} \) |
| 29 | \( 1 + (247. + 761. i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-1.48e3 + 4.55e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.10e4 + 8.02e3i)T + (2.14e7 - 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-2.61e3 + 1.90e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 + 1.89e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + (5.76e3 + 1.77e4i)T + (-1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-5.96e3 - 1.83e4i)T + (-3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (3.63e4 - 2.63e4i)T + (2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (-1.62e4 - 1.17e4i)T + (2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-1.37e4 + 4.23e4i)T + (-1.09e9 - 7.93e8i)T^{2} \) |
| 71 | \( 1 + (1.70e4 + 5.25e4i)T + (-1.45e9 + 1.06e9i)T^{2} \) |
| 73 | \( 1 + (3.03e4 + 2.20e4i)T + (6.40e8 + 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-4.17e3 - 1.28e4i)T + (-2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-5.50e3 + 1.69e4i)T + (-3.18e9 - 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-6.57e3 - 4.77e3i)T + (1.72e9 + 5.31e9i)T^{2} \) |
| 97 | \( 1 + (312. + 962. i)T + (-6.94e9 + 5.04e9i)T^{2} \) |
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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
−13.33116354247316647610287775858, −12.53453136530487460700185059343, −11.79880947145123585464658307397, −9.985535915564512532699179921078, −9.141043588639815741705846673724, −7.66269310693264264062777335186, −6.34668782858278258666082210918, −4.24579246784702159315437554418, −1.88572104150118141481348873065, −0.17743900212536647015472043854,
2.85581099686710929191150112048, 5.00613451103175315484626816132, 6.67603270067105607354591143519, 7.70231489094371920907370025880, 9.710984177514092666889263804729, 10.13123379486555094495175492769, 11.40983443195942861457779096415, 13.05184121345733467524370758830, 14.49744433334710592811496838837, 15.44867514028805380821393166461
