| L(s) = 1 | + (15.0 + 5.47i)2-s + (10.6 + 60.5i)3-s + (196. + 164. i)4-s + (1.28e3 − 1.07e3i)5-s + (−170. + 969. i)6-s + (−13.1 + 22.7i)7-s + (2.04e3 + 3.54e3i)8-s + (1.49e4 − 5.43e3i)9-s + (2.51e4 − 9.16e3i)10-s + (−1.63e4 − 2.82e4i)11-s + (−7.87e3 + 1.36e4i)12-s + (1.37e4 − 7.78e4i)13-s + (−321. + 269. i)14-s + (7.89e4 + 6.62e4i)15-s + (1.13e4 + 6.45e4i)16-s + (2.73e5 + 9.94e4i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.0761 + 0.431i)3-s + (0.383 + 0.321i)4-s + (0.918 − 0.770i)5-s + (−0.0538 + 0.305i)6-s + (−0.00206 + 0.00357i)7-s + (0.176 + 0.306i)8-s + (0.759 − 0.276i)9-s + (0.796 − 0.289i)10-s + (−0.335 − 0.581i)11-s + (−0.109 + 0.189i)12-s + (0.133 − 0.756i)13-s + (−0.00223 + 0.00187i)14-s + (0.402 + 0.337i)15-s + (0.0434 + 0.246i)16-s + (0.793 + 0.288i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.981 - 0.192i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(3.60148 + 0.349732i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.60148 + 0.349732i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-15.0 - 5.47i)T \) |
| 19 | \( 1 + (-3.12e5 - 4.74e5i)T \) |
| good | 3 | \( 1 + (-10.6 - 60.5i)T + (-1.84e4 + 6.73e3i)T^{2} \) |
| 5 | \( 1 + (-1.28e3 + 1.07e3i)T + (3.39e5 - 1.92e6i)T^{2} \) |
| 7 | \( 1 + (13.1 - 22.7i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (1.63e4 + 2.82e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (-1.37e4 + 7.78e4i)T + (-9.96e9 - 3.62e9i)T^{2} \) |
| 17 | \( 1 + (-2.73e5 - 9.94e4i)T + (9.08e10 + 7.62e10i)T^{2} \) |
| 23 | \( 1 + (-3.40e5 - 2.85e5i)T + (3.12e11 + 1.77e12i)T^{2} \) |
| 29 | \( 1 + (-1.94e6 + 7.08e5i)T + (1.11e13 - 9.32e12i)T^{2} \) |
| 31 | \( 1 + (2.79e5 - 4.84e5i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 - 2.80e6T + 1.29e14T^{2} \) |
| 41 | \( 1 + (2.71e6 + 1.53e7i)T + (-3.07e14 + 1.11e14i)T^{2} \) |
| 43 | \( 1 + (-1.39e6 + 1.16e6i)T + (8.72e13 - 4.94e14i)T^{2} \) |
| 47 | \( 1 + (3.83e7 - 1.39e7i)T + (8.57e14 - 7.19e14i)T^{2} \) |
| 53 | \( 1 + (4.17e7 + 3.50e7i)T + (5.72e14 + 3.24e15i)T^{2} \) |
| 59 | \( 1 + (1.47e8 + 5.36e7i)T + (6.63e15 + 5.56e15i)T^{2} \) |
| 61 | \( 1 + (1.20e8 + 1.01e8i)T + (2.03e15 + 1.15e16i)T^{2} \) |
| 67 | \( 1 + (1.17e8 - 4.25e7i)T + (2.08e16 - 1.74e16i)T^{2} \) |
| 71 | \( 1 + (1.56e6 - 1.30e6i)T + (7.96e15 - 4.51e16i)T^{2} \) |
| 73 | \( 1 + (-2.14e7 - 1.21e8i)T + (-5.53e16 + 2.01e16i)T^{2} \) |
| 79 | \( 1 + (-4.73e7 - 2.68e8i)T + (-1.12e17 + 4.09e16i)T^{2} \) |
| 83 | \( 1 + (-3.09e8 + 5.36e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (4.30e7 - 2.44e8i)T + (-3.29e17 - 1.19e17i)T^{2} \) |
| 97 | \( 1 + (-1.29e9 - 4.70e8i)T + (5.82e17 + 4.88e17i)T^{2} \) |
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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
−14.24160723654592609718527864872, −13.17760758351573033562586903069, −12.32680704048855508298883267810, −10.53773562694959130075185263198, −9.424717533599238093667326091173, −7.889770456446493505757295918623, −6.02471543893353590406634865531, −4.99641260961440421129840200450, −3.37416950161185974312889635544, −1.32538762354529629128703834709,
1.54816431563963027697447155760, 2.79351403397954886744976094829, 4.76620796450943872191161894008, 6.37347760101841626325146323335, 7.39351292735940473467535803265, 9.608209110375151417515397757880, 10.60278603495477721500877003725, 12.03509246232714836086676751304, 13.27722845566327326695264292204, 13.99067694116718927784007001835
