| L(s) = 1 | + (−4.36 + 24.7i)3-s + (−45.0 − 37.7i)5-s + (−12.6 − 21.9i)7-s + (−365. − 133. i)9-s + (4.84 − 8.39i)11-s + (−62.7 − 355. i)13-s + (1.13e3 − 949. i)15-s + (304. − 110. i)17-s + (1.53e3 − 350. i)19-s + (599. − 218. i)21-s + (1.52e3 − 1.27e3i)23-s + (57.2 + 324. i)25-s + (1.83e3 − 3.18e3i)27-s + (−5.87e3 − 2.13e3i)29-s + (−5.19e3 − 8.99e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.280 + 1.58i)3-s + (−0.805 − 0.675i)5-s + (−0.0978 − 0.169i)7-s + (−1.50 − 0.547i)9-s + (0.0120 − 0.0209i)11-s + (−0.102 − 0.584i)13-s + (1.29 − 1.09i)15-s + (0.255 − 0.0931i)17-s + (0.974 − 0.222i)19-s + (0.296 − 0.107i)21-s + (0.600 − 0.504i)23-s + (0.0183 + 0.103i)25-s + (0.485 − 0.840i)27-s + (−1.29 − 0.472i)29-s + (−0.970 − 1.68i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.459705 - 0.352067i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.459705 - 0.352067i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.53e3 + 350. i)T \) |
| good | 3 | \( 1 + (4.36 - 24.7i)T + (-228. - 83.1i)T^{2} \) |
| 5 | \( 1 + (45.0 + 37.7i)T + (542. + 3.07e3i)T^{2} \) |
| 7 | \( 1 + (12.6 + 21.9i)T + (-8.40e3 + 1.45e4i)T^{2} \) |
| 11 | \( 1 + (-4.84 + 8.39i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (62.7 + 355. i)T + (-3.48e5 + 1.26e5i)T^{2} \) |
| 17 | \( 1 + (-304. + 110. i)T + (1.08e6 - 9.12e5i)T^{2} \) |
| 23 | \( 1 + (-1.52e3 + 1.27e3i)T + (1.11e6 - 6.33e6i)T^{2} \) |
| 29 | \( 1 + (5.87e3 + 2.13e3i)T + (1.57e7 + 1.31e7i)T^{2} \) |
| 31 | \( 1 + (5.19e3 + 8.99e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 7.41e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (2.55e3 - 1.44e4i)T + (-1.08e8 - 3.96e7i)T^{2} \) |
| 43 | \( 1 + (-109. - 91.5i)T + (2.55e7 + 1.44e8i)T^{2} \) |
| 47 | \( 1 + (2.21e3 + 805. i)T + (1.75e8 + 1.47e8i)T^{2} \) |
| 53 | \( 1 + (-1.45e4 + 1.22e4i)T + (7.26e7 - 4.11e8i)T^{2} \) |
| 59 | \( 1 + (3.12e4 - 1.13e4i)T + (5.47e8 - 4.59e8i)T^{2} \) |
| 61 | \( 1 + (5.43e3 - 4.55e3i)T + (1.46e8 - 8.31e8i)T^{2} \) |
| 67 | \( 1 + (3.86e4 + 1.40e4i)T + (1.03e9 + 8.67e8i)T^{2} \) |
| 71 | \( 1 + (4.28e4 + 3.59e4i)T + (3.13e8 + 1.77e9i)T^{2} \) |
| 73 | \( 1 + (-5.41e3 + 3.07e4i)T + (-1.94e9 - 7.09e8i)T^{2} \) |
| 79 | \( 1 + (7.37e3 - 4.18e4i)T + (-2.89e9 - 1.05e9i)T^{2} \) |
| 83 | \( 1 + (-2.29e4 - 3.97e4i)T + (-1.96e9 + 3.41e9i)T^{2} \) |
| 89 | \( 1 + (3.62e3 + 2.05e4i)T + (-5.24e9 + 1.90e9i)T^{2} \) |
| 97 | \( 1 + (5.64e4 - 2.05e4i)T + (6.57e9 - 5.51e9i)T^{2} \) |
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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
−13.26558488093452873244600602492, −11.92073174949527754261114478092, −11.03500499061026892346121072424, −9.914283805028187558060207776124, −8.988036652949359822629969081537, −7.66702476426270814082515345193, −5.57520442664225753178900812996, −4.52308137549997817024988339075, −3.43801817174748477487181750010, −0.26690135082742827738621871076,
1.55184476781633227158138072728, 3.31075141125859492025472468693, 5.59299710291609174902089803034, 7.09318021829871492595079732661, 7.42887046537493387989775510192, 8.952181502353834030768844914726, 10.83212682946655321201208847832, 11.79772810075041284550014920819, 12.49142131443185329298470603805, 13.66615723466518282698899477812
