| L(s) = 1 | − 4·7-s + 13-s + 5·17-s + 6·19-s − 23-s − 5·25-s − 29-s + 6·37-s − 4·41-s − 43-s + 6·47-s + 9·49-s + 53-s + 4·59-s − 61-s + 10·71-s − 2·73-s − 3·79-s + 14·83-s − 4·91-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 20·119-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 0.277·13-s + 1.21·17-s + 1.37·19-s − 0.208·23-s − 25-s − 0.185·29-s + 0.986·37-s − 0.624·41-s − 0.152·43-s + 0.875·47-s + 9/7·49-s + 0.137·53-s + 0.520·59-s − 0.128·61-s + 1.18·71-s − 0.234·73-s − 0.337·79-s + 1.53·83-s − 0.419·91-s − 1.21·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 1.83·119-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 4 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
| 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
−13.28761542500111, −12.58381713791788, −12.33525629252626, −11.84352145292042, −11.43121983402815, −10.80965955998316, −10.18283902683267, −9.922103420975933, −9.412169620409273, −9.256676830581096, −8.429032531418749, −7.885125682843656, −7.547273619910381, −6.955241837533491, −6.469982051089845, −5.994305023131523, −5.457497190349627, −5.206804057382435, −4.167484099301685, −3.813054876598937, −3.281461092893725, −2.874758333142114, −2.227317866469202, −1.339980737755165, −0.7886701747709631, 0,
0.7886701747709631, 1.339980737755165, 2.227317866469202, 2.874758333142114, 3.281461092893725, 3.813054876598937, 4.167484099301685, 5.206804057382435, 5.457497190349627, 5.994305023131523, 6.469982051089845, 6.955241837533491, 7.547273619910381, 7.885125682843656, 8.429032531418749, 9.256676830581096, 9.412169620409273, 9.922103420975933, 10.18283902683267, 10.80965955998316, 11.43121983402815, 11.84352145292042, 12.33525629252626, 12.58381713791788, 13.28761542500111
