| L(s) = 1 | + (19.7 + 60.8i)2-s + (1.19e3 + 866. i)3-s + (−3.31e3 + 2.40e3i)4-s + (−1.62e4 + 4.99e4i)5-s + (−2.91e4 + 8.97e4i)6-s + (−2.03e5 + 1.47e5i)7-s + (−2.12e5 − 1.54e5i)8-s + (1.79e5 + 5.52e5i)9-s − 3.36e6·10-s + (−7.55e5 − 5.82e6i)11-s − 6.04e6·12-s + (2.64e6 + 8.13e6i)13-s + (−1.29e7 − 9.44e6i)14-s + (−6.26e7 + 4.55e7i)15-s + (5.18e6 − 1.59e7i)16-s + (1.00e7 − 3.09e7i)17-s + ⋯ |
| L(s) = 1 | + (0.218 + 0.672i)2-s + (0.944 + 0.686i)3-s + (−0.404 + 0.293i)4-s + (−0.464 + 1.42i)5-s + (−0.255 + 0.785i)6-s + (−0.652 + 0.474i)7-s + (−0.286 − 0.207i)8-s + (0.112 + 0.346i)9-s − 1.06·10-s + (−0.128 − 0.991i)11-s − 0.584·12-s + (0.151 + 0.467i)13-s + (−0.461 − 0.335i)14-s + (−1.42 + 1.03i)15-s + (0.0772 − 0.237i)16-s + (0.101 − 0.311i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (-0.757 + 0.652i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(1.654964908\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.654964908\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-19.7 - 60.8i)T \) |
| 11 | \( 1 + (7.55e5 + 5.82e6i)T \) |
| good | 3 | \( 1 + (-1.19e3 - 866. i)T + (4.92e5 + 1.51e6i)T^{2} \) |
| 5 | \( 1 + (1.62e4 - 4.99e4i)T + (-9.87e8 - 7.17e8i)T^{2} \) |
| 7 | \( 1 + (2.03e5 - 1.47e5i)T + (2.99e10 - 9.21e10i)T^{2} \) |
| 13 | \( 1 + (-2.64e6 - 8.13e6i)T + (-2.45e14 + 1.78e14i)T^{2} \) |
| 17 | \( 1 + (-1.00e7 + 3.09e7i)T + (-8.01e15 - 5.82e15i)T^{2} \) |
| 19 | \( 1 + (-2.79e8 - 2.02e8i)T + (1.29e16 + 3.99e16i)T^{2} \) |
| 23 | \( 1 + 2.93e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + (2.88e9 - 2.09e9i)T + (3.17e18 - 9.75e18i)T^{2} \) |
| 31 | \( 1 + (3.26e8 + 1.00e9i)T + (-1.97e19 + 1.43e19i)T^{2} \) |
| 37 | \( 1 + (4.20e9 - 3.05e9i)T + (7.52e19 - 2.31e20i)T^{2} \) |
| 41 | \( 1 + (2.43e10 + 1.77e10i)T + (2.85e20 + 8.79e20i)T^{2} \) |
| 43 | \( 1 + 6.71e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + (-7.64e10 - 5.55e10i)T + (1.68e21 + 5.19e21i)T^{2} \) |
| 53 | \( 1 + (-6.78e10 - 2.08e11i)T + (-2.10e22 + 1.53e22i)T^{2} \) |
| 59 | \( 1 + (-3.13e11 + 2.28e11i)T + (3.24e22 - 9.98e22i)T^{2} \) |
| 61 | \( 1 + (1.36e11 - 4.19e11i)T + (-1.30e23 - 9.51e22i)T^{2} \) |
| 67 | \( 1 + 1.03e12T + 5.48e23T^{2} \) |
| 71 | \( 1 + (6.07e11 - 1.86e12i)T + (-9.42e23 - 6.84e23i)T^{2} \) |
| 73 | \( 1 + (3.11e11 - 2.26e11i)T + (5.16e23 - 1.59e24i)T^{2} \) |
| 79 | \( 1 + (3.04e11 + 9.35e11i)T + (-3.77e24 + 2.74e24i)T^{2} \) |
| 83 | \( 1 + (-4.10e11 + 1.26e12i)T + (-7.17e24 - 5.21e24i)T^{2} \) |
| 89 | \( 1 - 1.62e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-3.05e12 - 9.39e12i)T + (-5.44e25 + 3.95e25i)T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.52097837415761155434518093541, −14.53384187749266485407948645087, −13.77722816188209064843406591509, −11.74208247691684275832909140543, −10.10661213084280514386338267608, −8.831783889549821927702768901795, −7.40379213668336355860704921039, −5.95726197856096274725124118757, −3.64503569489932124833072782418, −3.00290695997858487919666422350,
0.46059581780771040970308573645, 1.77699264427219907689491702731, 3.45016100409143803961530645710, 4.98877106454144206780554108531, 7.41904104274137910275167980343, 8.656164356417693667584251109739, 9.863740609190886174892613424734, 11.90305197104951954541509812397, 13.01258840736713622298333070223, 13.51484468698647899040264426243
