| L(s) = 1 | − 5-s + 7-s + 3·13-s + 7·17-s + 4·19-s + 8·23-s − 4·25-s − 29-s − 8·31-s − 35-s − 9·37-s + 10·41-s − 4·43-s + 8·47-s + 49-s − 10·53-s − 12·59-s − 5·61-s − 3·65-s + 12·67-s + 7·73-s − 8·79-s − 7·85-s + 7·89-s + 3·91-s − 4·95-s − 14·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.377·7-s + 0.832·13-s + 1.69·17-s + 0.917·19-s + 1.66·23-s − 4/5·25-s − 0.185·29-s − 1.43·31-s − 0.169·35-s − 1.47·37-s + 1.56·41-s − 0.609·43-s + 1.16·47-s + 1/7·49-s − 1.37·53-s − 1.56·59-s − 0.640·61-s − 0.372·65-s + 1.46·67-s + 0.819·73-s − 0.900·79-s − 0.759·85-s + 0.741·89-s + 0.314·91-s − 0.410·95-s − 1.42·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.104164449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.104164449\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 5 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.241928811235515392339750893462, −7.53659308822417846675325227384, −7.16979844216384808094492344955, −5.97171313858122670618588636861, −5.44632043638601605635246824540, −4.65820437680664781246242094245, −3.53970830630328683513448404035, −3.23147769773947596556088271605, −1.76712257236027183387518751705, −0.859762343924638168567215917639,
0.859762343924638168567215917639, 1.76712257236027183387518751705, 3.23147769773947596556088271605, 3.53970830630328683513448404035, 4.65820437680664781246242094245, 5.44632043638601605635246824540, 5.97171313858122670618588636861, 7.16979844216384808094492344955, 7.53659308822417846675325227384, 8.241928811235515392339750893462
