| L(s) = 1 | − 5-s + 2·7-s − 3·9-s + 8·19-s − 8·23-s + 25-s − 10·29-s + 8·31-s − 2·35-s − 10·37-s + 2·41-s + 6·43-s + 3·45-s − 8·47-s − 3·49-s + 14·53-s − 4·59-s − 10·61-s − 6·63-s + 4·67-s + 8·73-s + 4·79-s + 9·81-s − 10·83-s + 6·89-s − 8·95-s − 10·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 9-s + 1.83·19-s − 1.66·23-s + 1/5·25-s − 1.85·29-s + 1.43·31-s − 0.338·35-s − 1.64·37-s + 0.312·41-s + 0.914·43-s + 0.447·45-s − 1.16·47-s − 3/7·49-s + 1.92·53-s − 0.520·59-s − 1.28·61-s − 0.755·63-s + 0.488·67-s + 0.936·73-s + 0.450·79-s + 81-s − 1.09·83-s + 0.635·89-s − 0.820·95-s − 1.01·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 10 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
| 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
−7.891903022726509590950203031767, −7.44329852132488865055617834418, −6.42848486332572178429977240782, −5.55194915478733693744511488075, −5.14235249576602984209862498558, −4.07724055496098789531736361685, −3.38146488603572310878833868678, −2.43142573741392445437439718504, −1.36556677652435609395491816742, 0,
1.36556677652435609395491816742, 2.43142573741392445437439718504, 3.38146488603572310878833868678, 4.07724055496098789531736361685, 5.14235249576602984209862498558, 5.55194915478733693744511488075, 6.42848486332572178429977240782, 7.44329852132488865055617834418, 7.891903022726509590950203031767
