| L(s) = 1 | + (15.0 + 5.47i)2-s + (34.3 + 194. i)3-s + (196. + 164. i)4-s + (317. − 266. i)5-s + (−550. + 3.11e3i)6-s + (−2.69e3 + 4.66e3i)7-s + (2.04e3 + 3.54e3i)8-s + (−1.83e4 + 6.67e3i)9-s + (6.23e3 − 2.27e3i)10-s + (2.73e4 + 4.73e4i)11-s + (−2.53e4 + 4.38e4i)12-s + (−1.92e3 + 1.09e4i)13-s + (−6.59e4 + 5.53e4i)14-s + (6.29e4 + 5.28e4i)15-s + (1.13e4 + 6.45e4i)16-s + (−4.33e5 − 1.57e5i)17-s + ⋯ |
| L(s) = 1 | + (0.664 + 0.241i)2-s + (0.245 + 1.38i)3-s + (0.383 + 0.321i)4-s + (0.227 − 0.190i)5-s + (−0.173 + 0.982i)6-s + (−0.423 + 0.733i)7-s + (0.176 + 0.306i)8-s + (−0.931 + 0.339i)9-s + (0.197 − 0.0717i)10-s + (0.563 + 0.975i)11-s + (−0.352 + 0.611i)12-s + (−0.0186 + 0.106i)13-s + (−0.459 + 0.385i)14-s + (0.320 + 0.269i)15-s + (0.0434 + 0.246i)16-s + (−1.25 − 0.458i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.919 - 0.393i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(0.545361 + 2.65705i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.545361 + 2.65705i\) |
| \(L(\frac{11}{2})\) |
|
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 | \( 1 + (-9.56e3 + 5.67e5i)T \) |
| good | 3 | \( 1 + (-34.3 - 194. i)T + (-1.84e4 + 6.73e3i)T^{2} \) |
| 5 | \( 1 + (-317. + 266. i)T + (3.39e5 - 1.92e6i)T^{2} \) |
| 7 | \( 1 + (2.69e3 - 4.66e3i)T + (-2.01e7 - 3.49e7i)T^{2} \) |
| 11 | \( 1 + (-2.73e4 - 4.73e4i)T + (-1.17e9 + 2.04e9i)T^{2} \) |
| 13 | \( 1 + (1.92e3 - 1.09e4i)T + (-9.96e9 - 3.62e9i)T^{2} \) |
| 17 | \( 1 + (4.33e5 + 1.57e5i)T + (9.08e10 + 7.62e10i)T^{2} \) |
| 23 | \( 1 + (4.63e5 + 3.89e5i)T + (3.12e11 + 1.77e12i)T^{2} \) |
| 29 | \( 1 + (-2.79e6 + 1.01e6i)T + (1.11e13 - 9.32e12i)T^{2} \) |
| 31 | \( 1 + (6.60e5 - 1.14e6i)T + (-1.32e13 - 2.28e13i)T^{2} \) |
| 37 | \( 1 + 9.54e4T + 1.29e14T^{2} \) |
| 41 | \( 1 + (-1.40e6 - 7.96e6i)T + (-3.07e14 + 1.11e14i)T^{2} \) |
| 43 | \( 1 + (2.68e6 - 2.25e6i)T + (8.72e13 - 4.94e14i)T^{2} \) |
| 47 | \( 1 + (-1.52e7 + 5.55e6i)T + (8.57e14 - 7.19e14i)T^{2} \) |
| 53 | \( 1 + (-5.01e7 - 4.20e7i)T + (5.72e14 + 3.24e15i)T^{2} \) |
| 59 | \( 1 + (-1.35e8 - 4.92e7i)T + (6.63e15 + 5.56e15i)T^{2} \) |
| 61 | \( 1 + (5.88e7 + 4.93e7i)T + (2.03e15 + 1.15e16i)T^{2} \) |
| 67 | \( 1 + (-2.37e8 + 8.63e7i)T + (2.08e16 - 1.74e16i)T^{2} \) |
| 71 | \( 1 + (-2.55e8 + 2.14e8i)T + (7.96e15 - 4.51e16i)T^{2} \) |
| 73 | \( 1 + (-1.13e7 - 6.41e7i)T + (-5.53e16 + 2.01e16i)T^{2} \) |
| 79 | \( 1 + (-6.69e7 - 3.79e8i)T + (-1.12e17 + 4.09e16i)T^{2} \) |
| 83 | \( 1 + (5.86e7 - 1.01e8i)T + (-9.34e16 - 1.61e17i)T^{2} \) |
| 89 | \( 1 + (1.89e8 - 1.07e9i)T + (-3.29e17 - 1.19e17i)T^{2} \) |
| 97 | \( 1 + (1.45e8 + 5.30e7i)T + (5.82e17 + 4.88e17i)T^{2} \) |
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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
−15.20570342645131356532876560376, −13.83370375867128312887120489672, −12.50497814168915234214565913409, −11.18540531014046438386540870411, −9.693873345558863594783100891308, −8.889749910485406159133144651586, −6.75139995659404871347126140555, −5.12241814459250062006628550393, −4.12026949695438521562636561627, −2.52511836856171620319907160612,
0.76380703250793733270716893690, 2.17332929888897720732939856149, 3.79775072274424814420003227513, 6.08558564340501335921874058095, 6.93149051723696680804565047694, 8.394748742573508809403891270579, 10.31699352768677167050412812554, 11.68254398912117978607091229219, 12.83668538118547903955051986988, 13.65885477785313035989421578343
