| L(s) = 1 | + (19.7 − 60.8i)2-s + (−306. + 223. i)3-s + (−3.31e3 − 2.40e3i)4-s + (−5.17e3 − 1.59e4i)5-s + (7.50e3 + 2.30e4i)6-s + (2.91e5 + 2.11e5i)7-s + (−2.12e5 + 1.54e5i)8-s + (−4.48e5 + 1.37e6i)9-s − 1.07e6·10-s + (2.53e6 − 5.29e6i)11-s + 1.55e6·12-s + (−2.24e6 + 6.91e6i)13-s + (1.86e7 − 1.35e7i)14-s + (5.14e6 + 3.73e6i)15-s + (5.18e6 + 1.59e7i)16-s + (7.19e6 + 2.21e7i)17-s + ⋯ |
| L(s) = 1 | + (0.218 − 0.672i)2-s + (−0.243 + 0.176i)3-s + (−0.404 − 0.293i)4-s + (−0.148 − 0.456i)5-s + (0.0656 + 0.202i)6-s + (0.936 + 0.680i)7-s + (−0.286 + 0.207i)8-s + (−0.281 + 0.865i)9-s − 0.339·10-s + (0.431 − 0.901i)11-s + 0.150·12-s + (−0.129 + 0.397i)13-s + (0.662 − 0.481i)14-s + (0.116 + 0.0847i)15-s + (0.0772 + 0.237i)16-s + (0.0723 + 0.222i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & (0.515 + 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.063294073\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.063294073\) |
| \(L(\frac{15}{2})\) |
|
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 | \( 1 + (-2.53e6 + 5.29e6i)T \) |
| good | 3 | \( 1 + (306. - 223. i)T + (4.92e5 - 1.51e6i)T^{2} \) |
| 5 | \( 1 + (5.17e3 + 1.59e4i)T + (-9.87e8 + 7.17e8i)T^{2} \) |
| 7 | \( 1 + (-2.91e5 - 2.11e5i)T + (2.99e10 + 9.21e10i)T^{2} \) |
| 13 | \( 1 + (2.24e6 - 6.91e6i)T + (-2.45e14 - 1.78e14i)T^{2} \) |
| 17 | \( 1 + (-7.19e6 - 2.21e7i)T + (-8.01e15 + 5.82e15i)T^{2} \) |
| 19 | \( 1 + (-3.25e8 + 2.36e8i)T + (1.29e16 - 3.99e16i)T^{2} \) |
| 23 | \( 1 - 2.99e7T + 5.04e17T^{2} \) |
| 29 | \( 1 + (1.85e9 + 1.34e9i)T + (3.17e18 + 9.75e18i)T^{2} \) |
| 31 | \( 1 + (-2.30e9 + 7.08e9i)T + (-1.97e19 - 1.43e19i)T^{2} \) |
| 37 | \( 1 + (-1.67e10 - 1.21e10i)T + (7.52e19 + 2.31e20i)T^{2} \) |
| 41 | \( 1 + (-3.86e10 + 2.80e10i)T + (2.85e20 - 8.79e20i)T^{2} \) |
| 43 | \( 1 - 8.85e9T + 1.71e21T^{2} \) |
| 47 | \( 1 + (9.63e10 - 7.00e10i)T + (1.68e21 - 5.19e21i)T^{2} \) |
| 53 | \( 1 + (5.59e10 - 1.72e11i)T + (-2.10e22 - 1.53e22i)T^{2} \) |
| 59 | \( 1 + (-4.29e11 - 3.11e11i)T + (3.24e22 + 9.98e22i)T^{2} \) |
| 61 | \( 1 + (2.13e10 + 6.57e10i)T + (-1.30e23 + 9.51e22i)T^{2} \) |
| 67 | \( 1 - 6.63e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + (-4.27e11 - 1.31e12i)T + (-9.42e23 + 6.84e23i)T^{2} \) |
| 73 | \( 1 + (-5.16e11 - 3.74e11i)T + (5.16e23 + 1.59e24i)T^{2} \) |
| 79 | \( 1 + (-9.73e11 + 2.99e12i)T + (-3.77e24 - 2.74e24i)T^{2} \) |
| 83 | \( 1 + (9.67e10 + 2.97e11i)T + (-7.17e24 + 5.21e24i)T^{2} \) |
| 89 | \( 1 + 1.99e11T + 2.19e25T^{2} \) |
| 97 | \( 1 + (-1.77e12 + 5.47e12i)T + (-5.44e25 - 3.95e25i)T^{2} \) |
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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
−14.46464501207019662475115631103, −13.35606783644232897747500943013, −11.70757503872196432747286328641, −11.17191135827224048227350032614, −9.329738336201670698760203963774, −8.072746160415981620693785629952, −5.62901622952078490082015341445, −4.52324291826598085428187647812, −2.53880569585503031971410334308, −0.906284489657855779926924369229,
1.10335669890447671765798680410, 3.55258055173388150933075859145, 5.16108809553090833352078999671, 6.80795869623720756254488245886, 7.83305585812215289309824635657, 9.644860479923519511759146539855, 11.33039803122691573836915172633, 12.54499197932299064111941793042, 14.29225249099505968915300235757, 14.80636144111332575292303666812
