| L(s) = 1 | + (−26.9 − 1.88i)3-s + (43.3 + 25.0i)5-s + (47.2 + 81.8i)7-s + (721. + 101. i)9-s + (−300. + 173. i)11-s + (130. − 225. i)13-s + (−1.12e3 − 755. i)15-s − 1.80e3i·17-s − 8.10e3·19-s + (−1.11e3 − 2.29e3i)21-s + (−1.76e4 − 1.01e4i)23-s + (−6.55e3 − 1.13e4i)25-s + (−1.92e4 − 4.09e3i)27-s + (−6.41e3 + 3.70e3i)29-s + (4.89e3 − 8.47e3i)31-s + ⋯ |
| L(s) = 1 | + (−0.997 − 0.0698i)3-s + (0.346 + 0.200i)5-s + (0.137 + 0.238i)7-s + (0.990 + 0.139i)9-s + (−0.225 + 0.130i)11-s + (0.0592 − 0.102i)13-s + (−0.331 − 0.223i)15-s − 0.367i·17-s − 1.18·19-s + (−0.120 − 0.247i)21-s + (−1.44 − 0.836i)23-s + (−0.419 − 0.727i)25-s + (−0.978 − 0.208i)27-s + (−0.262 + 0.151i)29-s + (0.164 − 0.284i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0863212 - 0.301149i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0863212 - 0.301149i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (26.9 + 1.88i)T \) |
| good | 5 | \( 1 + (-43.3 - 25.0i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-47.2 - 81.8i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (300. - 173. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-130. + 225. i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 + 1.80e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 8.10e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (1.76e4 + 1.01e4i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (6.41e3 - 3.70e3i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (-4.89e3 + 8.47e3i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 2.22e3T + 2.56e9T^{2} \) |
| 41 | \( 1 + (1.17e4 + 6.75e3i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (3.27e4 + 5.67e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (8.08e4 - 4.66e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 - 8.46e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (1.08e5 + 6.25e4i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (-2.55e4 - 4.42e4i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (-9.91e4 + 1.71e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 3.97e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 5.22e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (2.73e5 + 4.73e5i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (6.42e5 - 3.70e5i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 + 2.67e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (7.02e5 + 1.21e6i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
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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
−12.77193103362009585863435036252, −11.88426739207177927302528586332, −10.71788319822116495034215266536, −9.863724182493136739266292702220, −8.217360874850594026323294241777, −6.71777837308479722314742627779, −5.71996892943020317591334436877, −4.33485824283076217493791955060, −2.07914225666757457812413304725, −0.13337228999483049185675344854,
1.64751345587999167515059141849, 4.05636142336779664278812123976, 5.44501691078157170914330856098, 6.51337868182257507782647396257, 7.978091547515624701332475861016, 9.587137947442024534844576090584, 10.59588942708195627796900284183, 11.60277134571283312236432577010, 12.71550893127555547085325347090, 13.65757243252264969160008505461
