L(s) = 1 | + (−14.4 − 81.9i)3-s + (267. − 224. i)5-s + (−728. + 1.26e3i)7-s + (−4.45e3 + 1.62e3i)9-s + (3.61e3 + 6.25e3i)11-s + (−1.58e3 + 8.96e3i)13-s + (−2.23e4 − 1.87e4i)15-s + (−1.12e4 − 4.08e3i)17-s + (−2.45e4 − 1.70e4i)19-s + (1.13e5 + 4.14e4i)21-s + (3.78e4 + 3.17e4i)23-s + (7.68e3 − 4.35e4i)25-s + (1.06e5 + 1.84e5i)27-s + (−7.96e4 + 2.89e4i)29-s + (−2.65e4 + 4.59e4i)31-s + ⋯ |
L(s) = 1 | + (−0.309 − 1.75i)3-s + (0.958 − 0.804i)5-s + (−0.802 + 1.39i)7-s + (−2.03 + 0.741i)9-s + (0.818 + 1.41i)11-s + (−0.199 + 1.13i)13-s + (−1.70 − 1.43i)15-s + (−0.554 − 0.201i)17-s + (−0.821 − 0.570i)19-s + (2.68 + 0.977i)21-s + (0.649 + 0.544i)23-s + (0.0983 − 0.557i)25-s + (1.03 + 1.79i)27-s + (−0.606 + 0.220i)29-s + (−0.160 + 0.277i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.602 - 0.798i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(4)\) |
\(\approx\) |
\(0.804747 + 0.401070i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.804747 + 0.401070i\) |
\(L(\frac{9}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 + (2.45e4 + 1.70e4i)T \) |
good | 3 | \( 1 + (14.4 + 81.9i)T + (-2.05e3 + 747. i)T^{2} \) |
| 5 | \( 1 + (-267. + 224. i)T + (1.35e4 - 7.69e4i)T^{2} \) |
| 7 | \( 1 + (728. - 1.26e3i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-3.61e3 - 6.25e3i)T + (-9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.58e3 - 8.96e3i)T + (-5.89e7 - 2.14e7i)T^{2} \) |
| 17 | \( 1 + (1.12e4 + 4.08e3i)T + (3.14e8 + 2.63e8i)T^{2} \) |
| 23 | \( 1 + (-3.78e4 - 3.17e4i)T + (5.91e8 + 3.35e9i)T^{2} \) |
| 29 | \( 1 + (7.96e4 - 2.89e4i)T + (1.32e10 - 1.10e10i)T^{2} \) |
| 31 | \( 1 + (2.65e4 - 4.59e4i)T + (-1.37e10 - 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.90e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.06e4 - 6.02e4i)T + (-1.83e11 + 6.66e10i)T^{2} \) |
| 43 | \( 1 + (-6.54e4 + 5.49e4i)T + (4.72e10 - 2.67e11i)T^{2} \) |
| 47 | \( 1 + (6.49e5 - 2.36e5i)T + (3.88e11 - 3.25e11i)T^{2} \) |
| 53 | \( 1 + (-7.59e5 - 6.36e5i)T + (2.03e11 + 1.15e12i)T^{2} \) |
| 59 | \( 1 + (-1.66e6 - 6.04e5i)T + (1.90e12 + 1.59e12i)T^{2} \) |
| 61 | \( 1 + (2.29e6 + 1.92e6i)T + (5.45e11 + 3.09e12i)T^{2} \) |
| 67 | \( 1 + (-2.00e6 + 7.30e5i)T + (4.64e12 - 3.89e12i)T^{2} \) |
| 71 | \( 1 + (-2.55e5 + 2.14e5i)T + (1.57e12 - 8.95e12i)T^{2} \) |
| 73 | \( 1 + (9.91e5 + 5.62e6i)T + (-1.03e13 + 3.77e12i)T^{2} \) |
| 79 | \( 1 + (-1.09e6 - 6.20e6i)T + (-1.80e13 + 6.56e12i)T^{2} \) |
| 83 | \( 1 + (4.00e6 - 6.93e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-8.14e5 + 4.62e6i)T + (-4.15e13 - 1.51e13i)T^{2} \) |
| 97 | \( 1 + (3.14e6 + 1.14e6i)T + (6.18e13 + 5.19e13i)T^{2} \) |
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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
−12.88258994042072883347137174148, −12.46722258122478829951328064730, −11.58684198074266908678338895840, −9.374065386812227394696263656510, −8.875992732015509663155218281452, −7.00528005002823465565876204343, −6.34713628747900986318323241747, −5.12774559585672869000556597331, −2.24359822631263042096126152109, −1.61397986182236203632479680368,
0.30173637108619030398314649081, 3.14924010324179686910356382759, 3.98898479391839707933282310445, 5.67353994261419998107963654517, 6.61180050795156277467971988264, 8.738508024929992634947646759373, 9.987079414800517616884072509725, 10.45480181394747598291979951668, 11.16484114829390073551558916784, 13.19630506276991384022980260111