L(s) = 1 | + (2.18 − 2.18i)3-s + (−1.79 + 1.79i)7-s − 6.58i·9-s + 0.913·11-s + (−1.73 − 1.73i)13-s + (−3 − 3i)17-s − 3.46i·19-s + 7.84i·21-s + (−3.79 − 3.79i)23-s + (−7.84 − 7.84i)27-s + 5.29·29-s − 7.58i·31-s + (1.99 − 1.99i)33-s + (−5.19 + 5.19i)37-s − 7.58·39-s + ⋯ |
L(s) = 1 | + (1.26 − 1.26i)3-s + (−0.677 + 0.677i)7-s − 2.19i·9-s + 0.275·11-s + (−0.480 − 0.480i)13-s + (−0.727 − 0.727i)17-s − 0.794i·19-s + 1.71i·21-s + (−0.790 − 0.790i)23-s + (−1.50 − 1.50i)27-s + 0.982·29-s − 1.36i·31-s + (0.348 − 0.348i)33-s + (−0.854 + 0.854i)37-s − 1.21·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.727 + 0.685i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.972147332\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.972147332\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + (-2.18 + 2.18i)T - 3iT^{2} \) |
| 7 | \( 1 + (1.79 - 1.79i)T - 7iT^{2} \) |
| 11 | \( 1 - 0.913T + 11T^{2} \) |
| 13 | \( 1 + (1.73 + 1.73i)T + 13iT^{2} \) |
| 17 | \( 1 + (3 + 3i)T + 17iT^{2} \) |
| 19 | \( 1 + 3.46iT - 19T^{2} \) |
| 23 | \( 1 + (3.79 + 3.79i)T + 23iT^{2} \) |
| 29 | \( 1 - 5.29T + 29T^{2} \) |
| 31 | \( 1 + 7.58iT - 31T^{2} \) |
| 37 | \( 1 + (5.19 - 5.19i)T - 37iT^{2} \) |
| 41 | \( 1 - 1.58T + 41T^{2} \) |
| 43 | \( 1 + (-0.361 + 0.361i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.79 + 3.79i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.64 - 2.64i)T + 53iT^{2} \) |
| 59 | \( 1 + 5.29iT - 59T^{2} \) |
| 61 | \( 1 + 6.20iT - 61T^{2} \) |
| 67 | \( 1 + (0.361 + 0.361i)T + 67iT^{2} \) |
| 71 | \( 1 + 4.41iT - 71T^{2} \) |
| 73 | \( 1 + (-8.58 + 8.58i)T - 73iT^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + (-3.10 + 3.10i)T - 83iT^{2} \) |
| 89 | \( 1 - 15.1iT - 89T^{2} \) |
| 97 | \( 1 + (-8.58 - 8.58i)T + 97iT^{2} \) |
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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
−9.118145262082130838688614479319, −8.235409793740451254297046833008, −7.64158293508869080125687689823, −6.64559181242840919985688277878, −6.33954456270885379243637060984, −4.95507264789396189869087507119, −3.66406965006767386993784660837, −2.64002934985062520897712609849, −2.24650422380676784134967462849, −0.61095944695128959076990092576,
1.87424781492639136427650538632, 3.01485099873229201490784399340, 3.87702192037804588662254131584, 4.28801762862615721521363946868, 5.42076989601649586303408966000, 6.61658050874274459835521856127, 7.47073805936899220178967510872, 8.386683739132600607113783980557, 8.964473661178827998658718271996, 9.728382120215094001341538674089