L(s) = 1 | − 5-s + 4·11-s + 13-s − 17-s + 4·19-s + 25-s + 2·29-s + 6·37-s + 6·41-s + 4·43-s − 7·49-s − 6·53-s − 4·55-s + 4·59-s + 6·61-s − 65-s − 12·67-s − 16·71-s − 6·73-s − 8·79-s + 12·83-s + 85-s − 2·89-s − 4·95-s + 2·97-s + 101-s + 103-s + ⋯ |
L(s) = 1 | − 0.447·5-s + 1.20·11-s + 0.277·13-s − 0.242·17-s + 0.917·19-s + 1/5·25-s + 0.371·29-s + 0.986·37-s + 0.937·41-s + 0.609·43-s − 49-s − 0.824·53-s − 0.539·55-s + 0.520·59-s + 0.768·61-s − 0.124·65-s − 1.46·67-s − 1.89·71-s − 0.702·73-s − 0.900·79-s + 1.31·83-s + 0.108·85-s − 0.211·89-s − 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 159120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 + T \) |
good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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
−13.42187975240044, −13.12969223944762, −12.49821228838734, −11.98809846898737, −11.69303853307512, −11.15702775481924, −10.86726407227865, −10.05782583447022, −9.722810902987563, −9.111168459854545, −8.823830948624291, −8.221651067782115, −7.613165114490871, −7.292518391794855, −6.677648505248490, −6.111477401528983, −5.813047858159107, −4.979479584416574, −4.428926329665892, −4.082994119861965, −3.375731189071332, −2.956516477856301, −2.201211178055238, −1.337519297013855, −0.9841385931023228, 0,
0.9841385931023228, 1.337519297013855, 2.201211178055238, 2.956516477856301, 3.375731189071332, 4.082994119861965, 4.428926329665892, 4.979479584416574, 5.813047858159107, 6.111477401528983, 6.677648505248490, 7.292518391794855, 7.613165114490871, 8.221651067782115, 8.823830948624291, 9.111168459854545, 9.722810902987563, 10.05782583447022, 10.86726407227865, 11.15702775481924, 11.69303853307512, 11.98809846898737, 12.49821228838734, 13.12969223944762, 13.42187975240044