| L(s) = 1 | − 2·5-s + 6·13-s + 17-s − 8·23-s − 25-s + 6·29-s − 8·31-s + 10·37-s − 6·41-s − 12·43-s + 10·53-s + 8·59-s − 6·61-s − 12·65-s − 12·67-s + 6·73-s + 8·79-s − 16·83-s − 2·85-s + 2·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + 16·115-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 1.66·13-s + 0.242·17-s − 1.66·23-s − 1/5·25-s + 1.11·29-s − 1.43·31-s + 1.64·37-s − 0.937·41-s − 1.82·43-s + 1.37·53-s + 1.04·59-s − 0.768·61-s − 1.48·65-s − 1.46·67-s + 0.702·73-s + 0.900·79-s − 1.75·83-s − 0.216·85-s + 0.211·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + 1.49·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 119952 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 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
−13.76442865680582, −13.27860669274940, −12.98513019181027, −12.12155898339269, −11.92271131062022, −11.42773948838140, −11.03149369724760, −10.35912162892314, −10.05690129506926, −9.402486679950182, −8.730863732818024, −8.308129887419286, −8.057866436484019, −7.416755338781729, −6.877695740933601, −6.205410192922492, −5.899182347817803, −5.266992520987224, −4.454659682833887, −4.059636022012423, −3.539303295212663, −3.134425473216619, −2.165041112327149, −1.576652452687475, −0.8013681837041943, 0,
0.8013681837041943, 1.576652452687475, 2.165041112327149, 3.134425473216619, 3.539303295212663, 4.059636022012423, 4.454659682833887, 5.266992520987224, 5.899182347817803, 6.205410192922492, 6.877695740933601, 7.416755338781729, 8.057866436484019, 8.308129887419286, 8.730863732818024, 9.402486679950182, 10.05690129506926, 10.35912162892314, 11.03149369724760, 11.42773948838140, 11.92271131062022, 12.12155898339269, 12.98513019181027, 13.27860669274940, 13.76442865680582
