L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s − 3·11-s − 12-s + 13-s + 14-s + 16-s − 18-s + 21-s + 3·22-s − 7·23-s + 24-s − 26-s − 27-s − 28-s + 4·29-s − 7·31-s − 32-s + 3·33-s + 36-s + 4·37-s − 39-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.904·11-s − 0.288·12-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.218·21-s + 0.639·22-s − 1.45·23-s + 0.204·24-s − 0.196·26-s − 0.192·27-s − 0.188·28-s + 0.742·29-s − 1.25·31-s − 0.176·32-s + 0.522·33-s + 1/6·36-s + 0.657·37-s − 0.160·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 379050 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 \) |
good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 7 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.46405285124509, −12.25845667292693, −11.84656415840430, −11.15675383776175, −10.82650719467325, −10.50693270742340, −10.04360461204860, −9.511722354365454, −9.269165839853911, −8.528415037894142, −8.147984122855463, −7.703157559582226, −7.297716848416997, −6.624188103698496, −6.385586252779724, −5.680811890803571, −5.475292319423966, −4.851569215597387, −4.084988458721117, −3.798451855811453, −3.019268608784466, −2.474919115575317, −1.976883919742489, −1.287139460118266, −0.5712621503573287, 0,
0.5712621503573287, 1.287139460118266, 1.976883919742489, 2.474919115575317, 3.019268608784466, 3.798451855811453, 4.084988458721117, 4.851569215597387, 5.475292319423966, 5.680811890803571, 6.385586252779724, 6.624188103698496, 7.297716848416997, 7.703157559582226, 8.147984122855463, 8.528415037894142, 9.269165839853911, 9.511722354365454, 10.04360461204860, 10.50693270742340, 10.82650719467325, 11.15675383776175, 11.84656415840430, 12.25845667292693, 12.46405285124509