| L(s) = 1 | + 1.73·2-s − 2·3-s + 0.999·4-s − 3.46·6-s + 3.46·7-s − 1.73·8-s + 9-s − 1.99·12-s + 5.99·14-s − 5·16-s − 6.92·17-s + 1.73·18-s + 6.92·19-s − 6.92·21-s − 6·23-s + 3.46·24-s + 4·27-s + 3.46·28-s + 4·31-s − 5.19·32-s − 11.9·34-s + 0.999·36-s − 10·37-s + 11.9·38-s − 6.92·41-s − 11.9·42-s − 3.46·43-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 1.15·3-s + 0.499·4-s − 1.41·6-s + 1.30·7-s − 0.612·8-s + 0.333·9-s − 0.577·12-s + 1.60·14-s − 1.25·16-s − 1.68·17-s + 0.408·18-s + 1.58·19-s − 1.51·21-s − 1.25·23-s + 0.707·24-s + 0.769·27-s + 0.654·28-s + 0.718·31-s − 0.918·32-s − 2.05·34-s + 0.166·36-s − 1.64·37-s + 1.94·38-s − 1.08·41-s − 1.85·42-s − 0.528·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 + 2T + 3T^{2} \) |
| 7 | \( 1 - 3.46T + 7T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 6.92T + 17T^{2} \) |
| 19 | \( 1 - 6.92T + 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 6.92T + 41T^{2} \) |
| 43 | \( 1 + 3.46T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 6T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.92T + 61T^{2} \) |
| 67 | \( 1 + 10T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 6.92T + 73T^{2} \) |
| 79 | \( 1 - 6.92T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 - 10T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.366233693642661372013880388151, −7.28084559401010097618895313607, −6.54471988288324646127848098086, −5.76380550638857076846614545834, −5.17437584526464953764901082216, −4.67889960416948878982598264478, −3.93988461365387891873046069700, −2.75626991463482279653623135928, −1.58699312365382491621906581844, 0,
1.58699312365382491621906581844, 2.75626991463482279653623135928, 3.93988461365387891873046069700, 4.67889960416948878982598264478, 5.17437584526464953764901082216, 5.76380550638857076846614545834, 6.54471988288324646127848098086, 7.28084559401010097618895313607, 8.366233693642661372013880388151
