L(s) = 1 | + (0.5 − 0.866i)5-s + (1.5 − 2.59i)9-s + (−2 − 3.46i)11-s + 2·13-s + (1 + 1.73i)17-s + (2 − 3.46i)19-s + (−2 + 3.46i)23-s + (−0.499 − 0.866i)25-s − 2·29-s + (−4 − 6.92i)31-s + (−3 + 5.19i)37-s + 6·41-s − 8·43-s + (−1.5 − 2.59i)45-s + (2 − 3.46i)47-s + ⋯ |
L(s) = 1 | + (0.223 − 0.387i)5-s + (0.5 − 0.866i)9-s + (−0.603 − 1.04i)11-s + 0.554·13-s + (0.242 + 0.420i)17-s + (0.458 − 0.794i)19-s + (−0.417 + 0.722i)23-s + (−0.0999 − 0.173i)25-s − 0.371·29-s + (−0.718 − 1.24i)31-s + (−0.493 + 0.854i)37-s + 0.937·41-s − 1.21·43-s + (−0.223 − 0.387i)45-s + (0.291 − 0.505i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.266 + 0.963i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.526149348\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.526149348\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.5 + 0.866i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (2 + 3.46i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-1 - 1.73i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2 + 3.46i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2 - 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3 - 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 + (-2 + 3.46i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2 + 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (3 + 5.19i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 16T + 83T^{2} \) |
| 89 | \( 1 + (3 - 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−9.031498475158104799975340742924, −8.202510110926725375561992325429, −7.48000776587592498954195944030, −6.42879345535061633993076438240, −5.81719052051386365643483156906, −4.98457729243109005876415650039, −3.84035859024125448052399271146, −3.18034617942723019355588467044, −1.75196843018542689560959330384, −0.55182041369798724864729292085,
1.55673831530387375016807690684, 2.44272373468884620665589966239, 3.57719903384142266783470345619, 4.61450515674435707832336479544, 5.34286416306667222857382590396, 6.23736011171599227422803098777, 7.31993189195403858431227493458, 7.58908477500992717469597102699, 8.610886997194145311513936571838, 9.512509086409007994572809483649