L(s) = 1 | − i·2-s − 0.618i·3-s − 4-s − 0.618·6-s − 1.61i·7-s + i·8-s + 2.61·9-s + 3.85·11-s + 0.618i·12-s + 4.09i·13-s − 1.61·14-s + 16-s + 5.09i·17-s − 2.61i·18-s + 4.85·19-s + ⋯ |
L(s) = 1 | − 0.707i·2-s − 0.356i·3-s − 0.5·4-s − 0.252·6-s − 0.611i·7-s + 0.353i·8-s + 0.872·9-s + 1.16·11-s + 0.178i·12-s + 1.13i·13-s − 0.432·14-s + 0.250·16-s + 1.23i·17-s − 0.617i·18-s + 1.11·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.813493148\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.813493148\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 23 | \( 1 - iT \) |
good | 3 | \( 1 + 0.618iT - 3T^{2} \) |
| 7 | \( 1 + 1.61iT - 7T^{2} \) |
| 11 | \( 1 - 3.85T + 11T^{2} \) |
| 13 | \( 1 - 4.09iT - 13T^{2} \) |
| 17 | \( 1 - 5.09iT - 17T^{2} \) |
| 19 | \( 1 - 4.85T + 19T^{2} \) |
| 29 | \( 1 - 4.76T + 29T^{2} \) |
| 31 | \( 1 + 2.09T + 31T^{2} \) |
| 37 | \( 1 - 2.47iT - 37T^{2} \) |
| 41 | \( 1 + 12.3T + 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 9.70iT - 47T^{2} \) |
| 53 | \( 1 + 8.47iT - 53T^{2} \) |
| 59 | \( 1 - 11.7T + 59T^{2} \) |
| 61 | \( 1 - 6.32T + 61T^{2} \) |
| 67 | \( 1 + 5.52iT - 67T^{2} \) |
| 71 | \( 1 - 7.09T + 71T^{2} \) |
| 73 | \( 1 + 1.23iT - 73T^{2} \) |
| 79 | \( 1 + 10.4T + 79T^{2} \) |
| 83 | \( 1 - 10.9iT - 83T^{2} \) |
| 89 | \( 1 - 1.52T + 89T^{2} \) |
| 97 | \( 1 + 14.6iT - 97T^{2} \) |
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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
−9.891100287966834553906229568108, −8.943498963343155621504905396376, −8.143139290951214611019996904129, −6.95848288594634006033874891950, −6.62215926578494399074139728032, −5.19638848581168430759171761565, −4.08420315522076697231963854073, −3.63443890204363162497164048483, −1.92255093531253886581469112916, −1.16448812169687817509094237970,
1.09598191234769499830773332591, 2.89349603939038432008604778389, 3.94534862451535870191446265769, 4.96907361019676594790498805106, 5.62172846952858172143661652394, 6.73230416182722236229586333079, 7.33577273748477952740903663362, 8.311198866374228622852868008555, 9.185219560691512855975838922228, 9.713189419846584181181313621733