| L(s) = 1 | + (−3.23 + 2.35i)2-s + (9.12 − 28.0i)3-s + (4.94 − 15.2i)4-s + (30.2 + 47.0i)5-s + (36.5 + 112. i)6-s + 204.·7-s + (19.7 + 60.8i)8-s + (−509. − 369. i)9-s + (−208. − 80.9i)10-s + (329. − 239. i)11-s + (−382. − 277. i)12-s + (−478. − 347. i)13-s + (−660. + 479. i)14-s + (1.59e3 − 421. i)15-s + (−207. − 150. i)16-s + (107. + 329. i)17-s + ⋯ |
| L(s) = 1 | + (−0.572 + 0.415i)2-s + (0.585 − 1.80i)3-s + (0.154 − 0.475i)4-s + (0.541 + 0.840i)5-s + (0.414 + 1.27i)6-s + 1.57·7-s + (0.109 + 0.336i)8-s + (−2.09 − 1.52i)9-s + (−0.659 − 0.255i)10-s + (0.820 − 0.595i)11-s + (−0.766 − 0.556i)12-s + (−0.786 − 0.571i)13-s + (−0.900 + 0.654i)14-s + (1.83 − 0.483i)15-s + (−0.202 − 0.146i)16-s + (0.0898 + 0.276i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.321 + 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.46322 - 1.04836i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.46322 - 1.04836i\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + (-30.2 - 47.0i)T \) |
| good | 3 | \( 1 + (-9.12 + 28.0i)T + (-196. - 142. i)T^{2} \) |
| 7 | \( 1 - 204.T + 1.68e4T^{2} \) |
| 11 | \( 1 + (-329. + 239. i)T + (4.97e4 - 1.53e5i)T^{2} \) |
| 13 | \( 1 + (478. + 347. i)T + (1.14e5 + 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-107. - 329. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (286. + 881. i)T + (-2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + (-251. + 182. i)T + (1.98e6 - 6.12e6i)T^{2} \) |
| 29 | \( 1 + (-147. + 454. i)T + (-1.65e7 - 1.20e7i)T^{2} \) |
| 31 | \( 1 + (-154. - 474. i)T + (-2.31e7 + 1.68e7i)T^{2} \) |
| 37 | \( 1 + (4.47e3 + 3.25e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-1.30e4 - 9.47e3i)T + (3.58e7 + 1.10e8i)T^{2} \) |
| 43 | \( 1 - 4.94e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + (-4.70e3 + 1.44e4i)T + (-1.85e8 - 1.34e8i)T^{2} \) |
| 53 | \( 1 + (1.19e4 - 3.67e4i)T + (-3.38e8 - 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-2.07e4 - 1.50e4i)T + (2.20e8 + 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.58e4 - 1.14e4i)T + (2.60e8 - 8.03e8i)T^{2} \) |
| 67 | \( 1 + (-1.55e4 - 4.80e4i)T + (-1.09e9 + 7.93e8i)T^{2} \) |
| 71 | \( 1 + (-7.74e3 + 2.38e4i)T + (-1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (2.96e4 - 2.15e4i)T + (6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-5.27e3 + 1.62e4i)T + (-2.48e9 - 1.80e9i)T^{2} \) |
| 83 | \( 1 + (-2.74e3 - 8.45e3i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 + (-1.12e5 + 8.18e4i)T + (1.72e9 - 5.31e9i)T^{2} \) |
| 97 | \( 1 + (4.67e4 - 1.43e5i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−14.44229080857385431568376366396, −13.49790525987994264119235626872, −11.97042793140709875082754480062, −10.92320934439239331728168463343, −8.986583118831279005222152085125, −7.909576209040989894258654749546, −7.07498010302456370560819371042, −5.84644609151593182466609450756, −2.46085583382631985394279107865, −1.19106736966427232797179241357,
1.99211686910923013881690392259, 4.20133693009669068423407069376, 5.07308387642462356938521643977, 8.045565903266133139031488916301, 9.076611738771306889071594876436, 9.766573075389562387826640179668, 10.95987416595850224118149520002, 12.05982778900143338053567604930, 14.11812394405399656294342855844, 14.69294129384727146784296236442
