| L(s) = 1 | + i·3-s − 1.58·5-s + (−2.37 − 1.15i)7-s − 9-s + 2.26·11-s − 0.548·13-s − 1.58i·15-s − 0.433i·17-s − 6.02i·19-s + (1.15 − 2.37i)21-s − 8.24i·23-s − 2.50·25-s − i·27-s − 0.548i·29-s − 7.50·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.706·5-s + (−0.899 − 0.436i)7-s − 0.333·9-s + 0.681·11-s − 0.152·13-s − 0.408i·15-s − 0.105i·17-s − 1.38i·19-s + (0.252 − 0.519i)21-s − 1.71i·23-s − 0.500·25-s − 0.192i·27-s − 0.101i·29-s − 1.34·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.109 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.437933 - 0.488774i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.437933 - 0.488774i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 + (2.37 + 1.15i)T \) |
| good | 5 | \( 1 + 1.58T + 5T^{2} \) |
| 11 | \( 1 - 2.26T + 11T^{2} \) |
| 13 | \( 1 + 0.548T + 13T^{2} \) |
| 17 | \( 1 + 0.433iT - 17T^{2} \) |
| 19 | \( 1 + 6.02iT - 19T^{2} \) |
| 23 | \( 1 + 8.24iT - 23T^{2} \) |
| 29 | \( 1 + 0.548iT - 29T^{2} \) |
| 31 | \( 1 + 7.50T + 31T^{2} \) |
| 37 | \( 1 + 4.21iT - 37T^{2} \) |
| 41 | \( 1 + 7.09iT - 41T^{2} \) |
| 43 | \( 1 - 1.82T + 43T^{2} \) |
| 47 | \( 1 - 11.5T + 47T^{2} \) |
| 53 | \( 1 + 3.71iT - 53T^{2} \) |
| 59 | \( 1 - 11.5iT - 59T^{2} \) |
| 61 | \( 1 + 12.1T + 61T^{2} \) |
| 67 | \( 1 + 9.35T + 67T^{2} \) |
| 71 | \( 1 + 1.27iT - 71T^{2} \) |
| 73 | \( 1 + 0.867iT - 73T^{2} \) |
| 79 | \( 1 - 10.3iT - 79T^{2} \) |
| 83 | \( 1 + 6.13iT - 83T^{2} \) |
| 89 | \( 1 - 7.95iT - 89T^{2} \) |
| 97 | \( 1 + 19.1iT - 97T^{2} \) |
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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
−10.41158234154214643930700048038, −9.246782756931326569240545524981, −8.868729615870129984272574967519, −7.51146325402263112446227491985, −6.83240565774184405648983506777, −5.77075263036239205342711268111, −4.43740640399668840209042788062, −3.83152387207771219405373674236, −2.65467301768831151414003556050, −0.34874024113885276657037637386,
1.62722858312280132427413613379, 3.20452624823094826529841821659, 3.98367166377610693448322644883, 5.56395358448841684168121169505, 6.27449670167474594735497548833, 7.33564270055649664704058936262, 7.949769636552029056143111104586, 9.074946994442595125718449369525, 9.698539284321317890797568281465, 10.84221007492470679025392376854
