| L(s) = 1 | + 2·5-s − 2·7-s + 2·17-s + 4·19-s + 4·23-s − 25-s − 6·29-s − 2·31-s − 4·35-s − 8·37-s − 2·41-s − 4·43-s − 12·47-s − 3·49-s − 6·53-s + 4·59-s − 12·67-s − 12·71-s − 6·73-s + 10·79-s − 16·83-s + 4·85-s + 10·89-s + 8·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.755·7-s + 0.485·17-s + 0.917·19-s + 0.834·23-s − 1/5·25-s − 1.11·29-s − 0.359·31-s − 0.676·35-s − 1.31·37-s − 0.312·41-s − 0.609·43-s − 1.75·47-s − 3/7·49-s − 0.824·53-s + 0.520·59-s − 1.46·67-s − 1.42·71-s − 0.702·73-s + 1.12·79-s − 1.75·83-s + 0.433·85-s + 1.05·89-s + 0.820·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 389376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9311309412\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9311309412\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−12.68337110841191, −11.86456087480079, −11.63835267620273, −11.17474155781388, −10.46758077125605, −10.11037019618110, −9.828515248893994, −9.324796662211432, −8.938898266944451, −8.537566076512273, −7.695427158851874, −7.478242528125365, −6.910400980697395, −6.373344992846798, −6.018303529933580, −5.501987194561516, −5.034446391551110, −4.675809902926487, −3.653867904362108, −3.441622897582064, −2.967225906139017, −2.290711447957333, −1.538658338583139, −1.372539343233146, −0.2337746778275941,
0.2337746778275941, 1.372539343233146, 1.538658338583139, 2.290711447957333, 2.967225906139017, 3.441622897582064, 3.653867904362108, 4.675809902926487, 5.034446391551110, 5.501987194561516, 6.018303529933580, 6.373344992846798, 6.910400980697395, 7.478242528125365, 7.695427158851874, 8.537566076512273, 8.938898266944451, 9.324796662211432, 9.828515248893994, 10.11037019618110, 10.46758077125605, 11.17474155781388, 11.63835267620273, 11.86456087480079, 12.68337110841191
