| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 21-s + 22-s − 23-s − 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 2-s − 3-s + 4-s − 5-s − 6-s + 7-s + 8-s + 9-s − 10-s + 11-s − 12-s + 13-s + 14-s + 15-s + 16-s + 17-s + 18-s + 19-s − 20-s − 21-s + 22-s − 23-s − 24-s + 25-s + 26-s − 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2297 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.907150292\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.907150292\) |
| \(L(1)\) |
\(\approx\) |
\(1.724246099\) |
| \(L(1)\) |
\(\approx\) |
\(1.724246099\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2297 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.002005319827275951452769483741, −18.84744257027808272409482621439, −18.238063897241178150680377815196, −17.362386121721264092457635189468, −16.42429271247478426385713803905, −16.14322415869036547134276018195, −15.34754707981643867138605187838, −14.506750200548341059925796259274, −14.02930035812992146137375765231, −12.95147442269205472275723453411, −12.11483713102594964208000526484, −11.7518145865128443338545153560, −11.21017330788259243635886127520, −10.598078748853392339511928535048, −9.52375123989722020108401589744, −8.14040010934318630547646692816, −7.6804059934607834080082800054, −6.78737554239312629688991263936, −6.0400232018998451039994492064, −5.29927794027987049006107849417, −4.48312018307902735606858875143, −3.9421266224076096638377510354, −3.15878150836863965265489117222, −1.57381857921262384380974836690, −1.08575860944583528884274541488,
1.08575860944583528884274541488, 1.57381857921262384380974836690, 3.15878150836863965265489117222, 3.9421266224076096638377510354, 4.48312018307902735606858875143, 5.29927794027987049006107849417, 6.0400232018998451039994492064, 6.78737554239312629688991263936, 7.6804059934607834080082800054, 8.14040010934318630547646692816, 9.52375123989722020108401589744, 10.598078748853392339511928535048, 11.21017330788259243635886127520, 11.7518145865128443338545153560, 12.11483713102594964208000526484, 12.95147442269205472275723453411, 14.02930035812992146137375765231, 14.506750200548341059925796259274, 15.34754707981643867138605187838, 16.14322415869036547134276018195, 16.42429271247478426385713803905, 17.362386121721264092457635189468, 18.238063897241178150680377815196, 18.84744257027808272409482621439, 20.002005319827275951452769483741
