| L(s) = 1 | + (71.5 − 26.0i)3-s + (66.9 − 379. i)5-s + (−231. − 401. i)7-s + (2.76e3 − 2.32e3i)9-s + (332. − 576. i)11-s + (−1.19e4 − 4.35e3i)13-s + (−5.09e3 − 2.88e4i)15-s + (−1.59e3 − 1.33e3i)17-s + (1.52e4 + 2.57e4i)19-s + (−2.70e4 − 2.26e4i)21-s + (1.21e4 + 6.86e4i)23-s + (−6.61e4 − 2.40e4i)25-s + (5.41e4 − 9.38e4i)27-s + (6.64e4 − 5.57e4i)29-s + (−5.64e3 − 9.77e3i)31-s + ⋯ |
| L(s) = 1 | + (1.52 − 0.556i)3-s + (0.239 − 1.35i)5-s + (−0.255 − 0.441i)7-s + (1.26 − 1.06i)9-s + (0.0753 − 0.130i)11-s + (−1.51 − 0.549i)13-s + (−0.389 − 2.21i)15-s + (−0.0786 − 0.0659i)17-s + (0.510 + 0.859i)19-s + (−0.636 − 0.534i)21-s + (0.207 + 1.17i)23-s + (−0.846 − 0.307i)25-s + (0.529 − 0.917i)27-s + (0.506 − 0.424i)29-s + (−0.0340 − 0.0589i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.52801 - 2.56549i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52801 - 2.56549i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-1.52e4 - 2.57e4i)T \) |
| good | 3 | \( 1 + (-71.5 + 26.0i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-66.9 + 379. i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (231. + 401. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-332. + 576. i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (1.19e4 + 4.35e3i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (1.59e3 + 1.33e3i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (-1.21e4 - 6.86e4i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (-6.64e4 + 5.57e4i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (5.64e3 + 9.77e3i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 1.17e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-7.43e5 + 2.70e5i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (-1.42e5 + 8.05e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (3.42e5 - 2.87e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (3.67e5 + 2.08e6i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (-9.29e5 - 7.79e5i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (-1.59e5 - 9.02e5i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (-7.70e5 + 6.46e5i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (3.91e5 - 2.22e6i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (-3.78e6 + 1.37e6i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (-2.29e6 + 8.35e5i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (-1.98e6 - 3.43e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (-6.72e6 - 2.44e6i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (1.01e7 + 8.51e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−12.90531544184277002569100815372, −12.12501744890072180741282773215, −9.943772058099753068082676312909, −9.192577948998048304468675962466, −8.115481629962132674024205267541, −7.27387676775704650053072245927, −5.27053460868585185241466959702, −3.69484078450403118334040621725, −2.18751904732829375716930328410, −0.830024389420878376018991580907,
2.44227597432455646519912535157, 2.93374482029343278651877356751, 4.55756442791428753818721942509, 6.64821634684507836263928912077, 7.69098758307174265049777697463, 9.137512026467895847415041172066, 9.811654760437392919858678481414, 10.92444100568420421593388163681, 12.49520432745705066177324265296, 13.90919141432105654128107329395
