| L(s) = 1 | − 5-s + 13-s + 6·17-s − 4·19-s + 25-s + 2·29-s + 8·31-s + 2·37-s + 6·41-s − 4·43-s − 8·47-s + 10·53-s + 4·59-s − 14·61-s − 65-s + 4·67-s − 12·71-s + 10·73-s + 8·79-s + 12·83-s − 6·85-s + 14·89-s + 4·95-s + 2·97-s + 101-s + 103-s + 107-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.277·13-s + 1.45·17-s − 0.917·19-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.328·37-s + 0.937·41-s − 0.609·43-s − 1.16·47-s + 1.37·53-s + 0.520·59-s − 1.79·61-s − 0.124·65-s + 0.488·67-s − 1.42·71-s + 1.17·73-s + 0.900·79-s + 1.31·83-s − 0.650·85-s + 1.48·89-s + 0.410·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−13.24665317140712, −12.50585607905678, −12.30509332074318, −11.76707524927366, −11.44732713606222, −10.73002777991655, −10.40634576534587, −10.03092850835901, −9.339836397759339, −9.031466631814791, −8.245591651593203, −8.037322682352058, −7.668436180273515, −6.935307287354950, −6.413127352383332, −6.139918067612036, −5.348617370389667, −5.002189037795509, −4.299917814499977, −3.918367710875544, −3.287276052367336, −2.784424767323495, −2.186972436223633, −1.313709980292491, −0.8771160115161669, 0,
0.8771160115161669, 1.313709980292491, 2.186972436223633, 2.784424767323495, 3.287276052367336, 3.918367710875544, 4.299917814499977, 5.002189037795509, 5.348617370389667, 6.139918067612036, 6.413127352383332, 6.935307287354950, 7.668436180273515, 8.037322682352058, 8.245591651593203, 9.031466631814791, 9.339836397759339, 10.03092850835901, 10.40634576534587, 10.73002777991655, 11.44732713606222, 11.76707524927366, 12.30509332074318, 12.50585607905678, 13.24665317140712
