| L(s) = 1 | + (−29.3 + 10.6i)3-s + (6.78 − 38.5i)5-s + (−102. − 176. i)7-s + (−929. + 779. i)9-s + (1.30e3 − 2.25e3i)11-s + (−3.81e3 − 1.38e3i)13-s + (211. + 1.20e3i)15-s + (474. + 397. i)17-s + (7.03e3 − 2.90e4i)19-s + (4.87e3 + 4.09e3i)21-s + (1.94e4 + 1.10e5i)23-s + (7.19e4 + 2.61e4i)25-s + (5.30e4 − 9.18e4i)27-s + (8.76e4 − 7.35e4i)29-s + (1.18e5 + 2.05e5i)31-s + ⋯ |
| L(s) = 1 | + (−0.627 + 0.228i)3-s + (0.0242 − 0.137i)5-s + (−0.112 − 0.194i)7-s + (−0.424 + 0.356i)9-s + (0.294 − 0.510i)11-s + (−0.481 − 0.175i)13-s + (0.0162 + 0.0919i)15-s + (0.0234 + 0.0196i)17-s + (0.235 − 0.971i)19-s + (0.114 + 0.0964i)21-s + (0.333 + 1.89i)23-s + (0.921 + 0.335i)25-s + (0.518 − 0.898i)27-s + (0.667 − 0.559i)29-s + (0.715 + 1.23i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 76 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.954 - 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.33822 + 0.203055i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33822 + 0.203055i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (-7.03e3 + 2.90e4i)T \) |
| good | 3 | \( 1 + (29.3 - 10.6i)T + (1.67e3 - 1.40e3i)T^{2} \) |
| 5 | \( 1 + (-6.78 + 38.5i)T + (-7.34e4 - 2.67e4i)T^{2} \) |
| 7 | \( 1 + (102. + 176. i)T + (-4.11e5 + 7.13e5i)T^{2} \) |
| 11 | \( 1 + (-1.30e3 + 2.25e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (3.81e3 + 1.38e3i)T + (4.80e7 + 4.03e7i)T^{2} \) |
| 17 | \( 1 + (-474. - 397. i)T + (7.12e7 + 4.04e8i)T^{2} \) |
| 23 | \( 1 + (-1.94e4 - 1.10e5i)T + (-3.19e9 + 1.16e9i)T^{2} \) |
| 29 | \( 1 + (-8.76e4 + 7.35e4i)T + (2.99e9 - 1.69e10i)T^{2} \) |
| 31 | \( 1 + (-1.18e5 - 2.05e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 - 4.29e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-2.06e5 + 7.52e4i)T + (1.49e11 - 1.25e11i)T^{2} \) |
| 43 | \( 1 + (6.67e4 - 3.78e5i)T + (-2.55e11 - 9.29e10i)T^{2} \) |
| 47 | \( 1 + (-1.69e5 + 1.42e5i)T + (8.79e10 - 4.98e11i)T^{2} \) |
| 53 | \( 1 + (-3.79e4 - 2.15e5i)T + (-1.10e12 + 4.01e11i)T^{2} \) |
| 59 | \( 1 + (-1.51e6 - 1.27e6i)T + (4.32e11 + 2.45e12i)T^{2} \) |
| 61 | \( 1 + (1.37e5 + 7.78e5i)T + (-2.95e12 + 1.07e12i)T^{2} \) |
| 67 | \( 1 + (1.47e6 - 1.23e6i)T + (1.05e12 - 5.96e12i)T^{2} \) |
| 71 | \( 1 + (-2.51e4 + 1.42e5i)T + (-8.54e12 - 3.11e12i)T^{2} \) |
| 73 | \( 1 + (-1.70e6 + 6.19e5i)T + (8.46e12 - 7.10e12i)T^{2} \) |
| 79 | \( 1 + (3.41e6 - 1.24e6i)T + (1.47e13 - 1.23e13i)T^{2} \) |
| 83 | \( 1 + (2.90e6 + 5.03e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (8.57e6 + 3.12e6i)T + (3.38e13 + 2.84e13i)T^{2} \) |
| 97 | \( 1 + (5.12e6 + 4.30e6i)T + (1.40e13 + 7.95e13i)T^{2} \) |
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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
−13.15696797956629941930787622214, −11.78561827988883979308844054613, −11.06189410520993778628966009366, −9.860385434779306913324725141085, −8.612231209687146993948919674836, −7.19245525179500368532641952401, −5.79169816216310663038606216039, −4.71697779612898911441555281811, −2.94680779879796880203266088340, −0.852451944877188447908540490578,
0.74896237168881213656905411231, 2.64822344663688198875202423617, 4.49240782263778767296124389647, 5.96023015159133124740797256219, 6.91354827611493793955644864327, 8.436996466662051096718528493430, 9.725086796538806420116560077527, 10.90180427731066102649175510571, 12.07408810157743884706398698766, 12.65209464179946585095035379479
