| L(s) = 1 | + (7.27 − 8.66i)2-s + (−36.0 + 72.5i)3-s + (−22.2 − 126. i)4-s + (0.862 − 2.37i)5-s + (366. + 839. i)6-s + (117. − 669. i)7-s + (−1.25e3 − 724. i)8-s + (−3.96e3 − 5.23e3i)9-s + (−14.2 − 24.7i)10-s + (3.73e3 + 1.02e4i)11-s + (9.94e3 + 2.93e3i)12-s + (−5.25e3 + 4.41e3i)13-s + (−4.94e3 − 5.88e3i)14-s + (140. + 148. i)15-s + (−1.53e4 + 5.60e3i)16-s + (9.61e4 − 5.54e4i)17-s + ⋯ |
| L(s) = 1 | + (0.454 − 0.541i)2-s + (−0.445 + 0.895i)3-s + (−0.0868 − 0.492i)4-s + (0.00138 − 0.00379i)5-s + (0.282 + 0.648i)6-s + (0.0491 − 0.278i)7-s + (−0.306 − 0.176i)8-s + (−0.603 − 0.797i)9-s + (−0.00142 − 0.00247i)10-s + (0.254 + 0.700i)11-s + (0.479 + 0.141i)12-s + (−0.184 + 0.154i)13-s + (−0.128 − 0.153i)14-s + (0.00278 + 0.00292i)15-s + (−0.234 + 0.0855i)16-s + (1.15 − 0.664i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.607 + 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.77231 - 0.876390i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.77231 - 0.876390i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-7.27 + 8.66i)T \) |
| 3 | \( 1 + (36.0 - 72.5i)T \) |
| good | 5 | \( 1 + (-0.862 + 2.37i)T + (-2.99e5 - 2.51e5i)T^{2} \) |
| 7 | \( 1 + (-117. + 669. i)T + (-5.41e6 - 1.97e6i)T^{2} \) |
| 11 | \( 1 + (-3.73e3 - 1.02e4i)T + (-1.64e8 + 1.37e8i)T^{2} \) |
| 13 | \( 1 + (5.25e3 - 4.41e3i)T + (1.41e8 - 8.03e8i)T^{2} \) |
| 17 | \( 1 + (-9.61e4 + 5.54e4i)T + (3.48e9 - 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-1.20e5 + 2.09e5i)T + (-8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (-9.71e4 + 1.71e4i)T + (7.35e10 - 2.67e10i)T^{2} \) |
| 29 | \( 1 + (-3.29e5 + 3.92e5i)T + (-8.68e10 - 4.92e11i)T^{2} \) |
| 31 | \( 1 + (2.39e5 + 1.35e6i)T + (-8.01e11 + 2.91e11i)T^{2} \) |
| 37 | \( 1 + (-6.30e5 - 1.09e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + (1.77e6 + 2.11e6i)T + (-1.38e12 + 7.86e12i)T^{2} \) |
| 43 | \( 1 + (1.23e5 - 4.48e4i)T + (8.95e12 - 7.51e12i)T^{2} \) |
| 47 | \( 1 + (-1.64e6 - 2.89e5i)T + (2.23e13 + 8.14e12i)T^{2} \) |
| 53 | \( 1 + 6.15e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + (9.01e5 - 2.47e6i)T + (-1.12e14 - 9.43e13i)T^{2} \) |
| 61 | \( 1 + (3.36e5 - 1.90e6i)T + (-1.80e14 - 6.55e13i)T^{2} \) |
| 67 | \( 1 + (1.28e7 - 1.07e7i)T + (7.05e13 - 3.99e14i)T^{2} \) |
| 71 | \( 1 + (3.26e6 - 1.88e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (1.56e7 - 2.71e7i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (1.69e7 + 1.41e7i)T + (2.63e14 + 1.49e15i)T^{2} \) |
| 83 | \( 1 + (-1.89e7 + 2.26e7i)T + (-3.91e14 - 2.21e15i)T^{2} \) |
| 89 | \( 1 + (-1.93e7 - 1.11e7i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (9.38e7 - 3.41e7i)T + (6.00e15 - 5.03e15i)T^{2} \) |
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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
−13.45949805058589343058100893956, −12.04334562207179142497449113551, −11.26985322081304057452690532294, −10.03053898156322007065172863435, −9.215674868689897486537272719855, −7.10028883742512344221767276223, −5.41322606678106994503994107948, −4.39977403609110820708820087103, −2.94618311867489712991654687113, −0.75385045735363171722518786766,
1.23099644412416907325072612883, 3.22680060865903575897799382847, 5.28655266714776007388777527677, 6.21436292847471236857966738997, 7.54713182224350653175589677557, 8.571653697953031019591146949699, 10.49659962295683198099281514593, 11.98558355249820981935759155540, 12.55744844687418902099979777333, 13.91932041362423918350778393296
