| L(s) = 1 | + (1.53 + 1.28i)2-s + (2.81 − 1.02i)3-s + (0.694 + 3.93i)4-s + (−1.98 + 11.2i)5-s + (5.63 + 2.05i)6-s + (4.45 + 7.71i)7-s + (−4.00 + 6.92i)8-s + (6.89 − 5.78i)9-s + (−17.4 + 14.6i)10-s + (−11.2 + 19.5i)11-s + (6 + 10.3i)12-s + (10.8 + 3.93i)13-s + (−3.09 + 17.5i)14-s + (5.94 + 33.7i)15-s + (−15.0 + 5.47i)16-s + (24.2 + 20.3i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (0.542 − 0.197i)3-s + (0.0868 + 0.492i)4-s + (−0.177 + 1.00i)5-s + (0.383 + 0.139i)6-s + (0.240 + 0.416i)7-s + (−0.176 + 0.306i)8-s + (0.255 − 0.214i)9-s + (−0.552 + 0.463i)10-s + (−0.309 + 0.536i)11-s + (0.144 + 0.249i)12-s + (0.230 + 0.0838i)13-s + (−0.0590 + 0.334i)14-s + (0.102 + 0.580i)15-s + (−0.234 + 0.0855i)16-s + (0.345 + 0.290i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0971 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.0971 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.79716 + 1.63025i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.79716 + 1.63025i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.53 - 1.28i)T \) |
| 3 | \( 1 + (-2.81 + 1.02i)T \) |
| 19 | \( 1 + (-75.9 + 33.0i)T \) |
| good | 5 | \( 1 + (1.98 - 11.2i)T + (-117. - 42.7i)T^{2} \) |
| 7 | \( 1 + (-4.45 - 7.71i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (11.2 - 19.5i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-10.8 - 3.93i)T + (1.68e3 + 1.41e3i)T^{2} \) |
| 17 | \( 1 + (-24.2 - 20.3i)T + (853. + 4.83e3i)T^{2} \) |
| 23 | \( 1 + (8.61 + 48.8i)T + (-1.14e4 + 4.16e3i)T^{2} \) |
| 29 | \( 1 + (18.2 - 15.3i)T + (4.23e3 - 2.40e4i)T^{2} \) |
| 31 | \( 1 + (77.5 + 134. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 88.9T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-1.65 + 0.601i)T + (5.27e4 - 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-87.2 + 494. i)T + (-7.47e4 - 2.71e4i)T^{2} \) |
| 47 | \( 1 + (-136. + 114. i)T + (1.80e4 - 1.02e5i)T^{2} \) |
| 53 | \( 1 + (3.89 + 22.0i)T + (-1.39e5 + 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-462. - 388. i)T + (3.56e4 + 2.02e5i)T^{2} \) |
| 61 | \( 1 + (119. + 678. i)T + (-2.13e5 + 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-283. + 238. i)T + (5.22e4 - 2.96e5i)T^{2} \) |
| 71 | \( 1 + (98.9 - 561. i)T + (-3.36e5 - 1.22e5i)T^{2} \) |
| 73 | \( 1 + (645. - 234. i)T + (2.98e5 - 2.50e5i)T^{2} \) |
| 79 | \( 1 + (574. - 208. i)T + (3.77e5 - 3.16e5i)T^{2} \) |
| 83 | \( 1 + (-312. - 541. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-1.88 - 0.687i)T + (5.40e5 + 4.53e5i)T^{2} \) |
| 97 | \( 1 + (-136. - 114. i)T + (1.58e5 + 8.98e5i)T^{2} \) |
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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
−13.56502512970190817156592594664, −12.46968822317699633256869700776, −11.44146756898850730701431183149, −10.22864486826566458938573633055, −8.838433622123320554976029483788, −7.62278761313640028286323967851, −6.82423852692093994169373279872, −5.40155621304756021421146872306, −3.73712235427913709663603012776, −2.42920125698750703651280776559,
1.19519208930615013224118983392, 3.21107372995545555501890451810, 4.50763172188920819786873207451, 5.62314721846035616703839913681, 7.51210428891379085443744267793, 8.629028407194727186162222615199, 9.698386818869541770287713664239, 10.85815682564346984957270816214, 11.96712389746839292463935578448, 12.95992255598846620715796948990
