| L(s) = 1 | + (1.61 − 0.618i)3-s − 5-s + (−0.381 − 2.61i)7-s + (2.23 − 2.00i)9-s − 0.763i·11-s − 1.23i·13-s + (−1.61 + 0.618i)15-s + 4.47·17-s − 7.23i·19-s + (−2.23 − 4i)21-s + 7.70i·23-s + 25-s + (2.38 − 4.61i)27-s + 4i·29-s + 3.23i·31-s + ⋯ |
| L(s) = 1 | + (0.934 − 0.356i)3-s − 0.447·5-s + (−0.144 − 0.989i)7-s + (0.745 − 0.666i)9-s − 0.230i·11-s − 0.342i·13-s + (−0.417 + 0.159i)15-s + 1.08·17-s − 1.66i·19-s + (−0.487 − 0.872i)21-s + 1.60i·23-s + 0.200·25-s + (0.458 − 0.888i)27-s + 0.742i·29-s + 0.581i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 420 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.487 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49136 - 0.874872i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49136 - 0.874872i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.61 + 0.618i)T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + (0.381 + 2.61i)T \) |
| good | 11 | \( 1 + 0.763iT - 11T^{2} \) |
| 13 | \( 1 + 1.23iT - 13T^{2} \) |
| 17 | \( 1 - 4.47T + 17T^{2} \) |
| 19 | \( 1 + 7.23iT - 19T^{2} \) |
| 23 | \( 1 - 7.70iT - 23T^{2} \) |
| 29 | \( 1 - 4iT - 29T^{2} \) |
| 31 | \( 1 - 3.23iT - 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 + 3.52T + 41T^{2} \) |
| 43 | \( 1 + 7.23T + 43T^{2} \) |
| 47 | \( 1 + 3.23T + 47T^{2} \) |
| 53 | \( 1 - 9.23iT - 53T^{2} \) |
| 59 | \( 1 - 8.94T + 59T^{2} \) |
| 61 | \( 1 - 4.94iT - 61T^{2} \) |
| 67 | \( 1 - 9.70T + 67T^{2} \) |
| 71 | \( 1 + 12.1iT - 71T^{2} \) |
| 73 | \( 1 - 6.76iT - 73T^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + 7.23T + 83T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 - 14.1iT - 97T^{2} \) |
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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
−11.02849629133706011221133314264, −10.01915043088126566368736440748, −9.203613190867104347839942612715, −8.152504848560974395356514449974, −7.41859922603523780286884010778, −6.71868273291628556324694221350, −5.10765887403387151069171293952, −3.78819398545219147696692162072, −3.01126012207753158160651467193, −1.13345449806548433860708418554,
2.04547161864513691826590735239, 3.26502875099632618278479471571, 4.30005880121403821981608870605, 5.52696253954270562868968807154, 6.75684513321432908225229995446, 8.144072270068138987658949031568, 8.330418892843298235734525655700, 9.667534837722314242501623445274, 10.08793569034412982533572132506, 11.42733813003976627487873369726
