| L(s) = 1 | + 5-s + 2·7-s − 13-s − 6.47·17-s − 2·19-s − 2.47·23-s + 25-s + 6.47·29-s − 4.47·31-s + 2·35-s − 6.94·37-s + 10.9·41-s + 4·47-s − 3·49-s − 8.94·53-s + 12.9·59-s + 6.94·61-s − 65-s + 3.52·67-s + 8.94·71-s − 3.52·73-s + 16.9·79-s + 8·83-s − 6.47·85-s + 2·89-s − 2·91-s − 2·95-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 0.755·7-s − 0.277·13-s − 1.56·17-s − 0.458·19-s − 0.515·23-s + 0.200·25-s + 1.20·29-s − 0.803·31-s + 0.338·35-s − 1.14·37-s + 1.70·41-s + 0.583·47-s − 0.428·49-s − 1.22·53-s + 1.68·59-s + 0.889·61-s − 0.124·65-s + 0.430·67-s + 1.06·71-s − 0.412·73-s + 1.90·79-s + 0.878·83-s − 0.702·85-s + 0.211·89-s − 0.209·91-s − 0.205·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.170175652\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.170175652\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 6.47T + 17T^{2} \) |
| 19 | \( 1 + 2T + 19T^{2} \) |
| 23 | \( 1 + 2.47T + 23T^{2} \) |
| 29 | \( 1 - 6.47T + 29T^{2} \) |
| 31 | \( 1 + 4.47T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 - 10.9T + 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 4T + 47T^{2} \) |
| 53 | \( 1 + 8.94T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 - 6.94T + 61T^{2} \) |
| 67 | \( 1 - 3.52T + 67T^{2} \) |
| 71 | \( 1 - 8.94T + 71T^{2} \) |
| 73 | \( 1 + 3.52T + 73T^{2} \) |
| 79 | \( 1 - 16.9T + 79T^{2} \) |
| 83 | \( 1 - 8T + 83T^{2} \) |
| 89 | \( 1 - 2T + 89T^{2} \) |
| 97 | \( 1 - 4.47T + 97T^{2} \) |
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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
−7.77178527368522457526115567791, −6.91007630187681357623891050701, −6.43717060689199147884568150547, −5.62788770923469182055400981431, −4.86702131929539501683163360965, −4.37721376004641010155950811242, −3.48311784552386331375692205206, −2.29287684528568941328194326822, −1.99159583294519917156577722064, −0.69218350628096647705185835349,
0.69218350628096647705185835349, 1.99159583294519917156577722064, 2.29287684528568941328194326822, 3.48311784552386331375692205206, 4.37721376004641010155950811242, 4.86702131929539501683163360965, 5.62788770923469182055400981431, 6.43717060689199147884568150547, 6.91007630187681357623891050701, 7.77178527368522457526115567791
