| L(s) = 1 | + (−10.5 − 7.65i)2-s − 51.6·3-s + (12.8 + 39.4i)4-s + (−26.1 − 80.5i)5-s + (543. + 395. i)6-s + (−1.17e3 + 856. i)7-s + (−347. + 1.07e3i)8-s + 478.·9-s + (−340. + 1.04e3i)10-s + (−609. + 1.87e3i)11-s + (−662. − 2.03e3i)12-s + (−1.94e3 − 1.41e3i)13-s + 1.89e4·14-s + (1.35e3 + 4.15e3i)15-s + (1.61e4 − 1.17e4i)16-s + (1.16e4 − 3.59e4i)17-s + ⋯ |
| L(s) = 1 | + (−0.930 − 0.676i)2-s − 1.10·3-s + (0.100 + 0.308i)4-s + (−0.0936 − 0.288i)5-s + (1.02 + 0.746i)6-s + (−1.29 + 0.944i)7-s + (−0.240 + 0.739i)8-s + 0.218·9-s + (−0.107 + 0.331i)10-s + (−0.138 + 0.425i)11-s + (−0.110 − 0.340i)12-s + (−0.245 − 0.178i)13-s + 1.84·14-s + (0.103 + 0.318i)15-s + (0.986 − 0.716i)16-s + (0.575 − 1.77i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.253643 - 0.112813i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.253643 - 0.112813i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 + (2.32e5 - 3.75e5i)T \) |
| good | 2 | \( 1 + (10.5 + 7.65i)T + (39.5 + 121. i)T^{2} \) |
| 3 | \( 1 + 51.6T + 2.18e3T^{2} \) |
| 5 | \( 1 + (26.1 + 80.5i)T + (-6.32e4 + 4.59e4i)T^{2} \) |
| 7 | \( 1 + (1.17e3 - 856. i)T + (2.54e5 - 7.83e5i)T^{2} \) |
| 11 | \( 1 + (609. - 1.87e3i)T + (-1.57e7 - 1.14e7i)T^{2} \) |
| 13 | \( 1 + (1.94e3 + 1.41e3i)T + (1.93e7 + 5.96e7i)T^{2} \) |
| 17 | \( 1 + (-1.16e4 + 3.59e4i)T + (-3.31e8 - 2.41e8i)T^{2} \) |
| 19 | \( 1 + (4.19e4 - 3.04e4i)T + (2.76e8 - 8.50e8i)T^{2} \) |
| 23 | \( 1 + (7.96e4 + 5.78e4i)T + (1.05e9 + 3.23e9i)T^{2} \) |
| 29 | \( 1 + (-6.92e4 - 2.13e5i)T + (-1.39e10 + 1.01e10i)T^{2} \) |
| 31 | \( 1 + (-2.67e4 + 8.23e4i)T + (-2.22e10 - 1.61e10i)T^{2} \) |
| 37 | \( 1 + (5.16e4 + 1.58e5i)T + (-7.68e10 + 5.57e10i)T^{2} \) |
| 43 | \( 1 + (1.68e5 + 1.22e5i)T + (8.39e10 + 2.58e11i)T^{2} \) |
| 47 | \( 1 + (-2.66e5 - 1.93e5i)T + (1.56e11 + 4.81e11i)T^{2} \) |
| 53 | \( 1 + (-8.71e4 - 2.68e5i)T + (-9.50e11 + 6.90e11i)T^{2} \) |
| 59 | \( 1 + (-1.22e6 - 8.92e5i)T + (7.69e11 + 2.36e12i)T^{2} \) |
| 61 | \( 1 + (-7.45e5 + 5.41e5i)T + (9.71e11 - 2.98e12i)T^{2} \) |
| 67 | \( 1 + (1.73e5 + 5.32e5i)T + (-4.90e12 + 3.56e12i)T^{2} \) |
| 71 | \( 1 + (4.12e5 - 1.26e6i)T + (-7.35e12 - 5.34e12i)T^{2} \) |
| 73 | \( 1 - 4.79e6T + 1.10e13T^{2} \) |
| 79 | \( 1 - 2.52e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 7.62e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (5.32e6 - 3.86e6i)T + (1.36e13 - 4.20e13i)T^{2} \) |
| 97 | \( 1 + (2.42e6 + 7.45e6i)T + (-6.53e13 + 4.74e13i)T^{2} \) |
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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
−14.50570301745214347414805876428, −12.39231897616207676931491937302, −12.09987570311379486681397252517, −10.58870837420573998396975647503, −9.723897207613841002830209754394, −8.517614983547878852328680218770, −6.41743718269703104480909163580, −5.21145004005615091270494441669, −2.57704373676173091908250466429, −0.45912566106010427820260452753,
0.46528417516899395788591212785, 3.75950340224015768912995803303, 6.12029688017104714500020257861, 6.81262224793921799105731217015, 8.308696095179776964135406505790, 9.925122621871831173588401640449, 10.71600565066606367482945567748, 12.30783828929845297955587924415, 13.40274365989321975796650815642, 15.26868377696305164563309590082
