| L(s) = 1 | + 5-s − 2·7-s − 11-s + 5.46·13-s − 5.46·19-s + 6.92·23-s + 25-s + 3.46·29-s + 10.9·31-s − 2·35-s − 4.92·37-s − 3.46·41-s + 4.92·43-s − 6.92·47-s − 3·49-s − 0.928·53-s − 55-s − 6.92·59-s + 2·61-s + 5.46·65-s − 8·67-s + 13.8·71-s − 8.39·73-s + 2·77-s + 6.53·79-s + 8.53·83-s − 0.928·89-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.755·7-s − 0.301·11-s + 1.51·13-s − 1.25·19-s + 1.44·23-s + 0.200·25-s + 0.643·29-s + 1.96·31-s − 0.338·35-s − 0.810·37-s − 0.541·41-s + 0.751·43-s − 1.01·47-s − 0.428·49-s − 0.127·53-s − 0.134·55-s − 0.901·59-s + 0.256·61-s + 0.677·65-s − 0.977·67-s + 1.64·71-s − 0.982·73-s + 0.227·77-s + 0.735·79-s + 0.936·83-s − 0.0983·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.095451468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.095451468\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 7 | \( 1 + 2T + 7T^{2} \) |
| 13 | \( 1 - 5.46T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 5.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 10.9T + 31T^{2} \) |
| 37 | \( 1 + 4.92T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 + 6.92T + 47T^{2} \) |
| 53 | \( 1 + 0.928T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 + 8.39T + 73T^{2} \) |
| 79 | \( 1 - 6.53T + 79T^{2} \) |
| 83 | \( 1 - 8.53T + 83T^{2} \) |
| 89 | \( 1 + 0.928T + 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.026407708054951337820521172458, −6.85420855449600597353886994838, −6.47747047049014084290698821884, −5.96454399954361730152367859326, −5.01986966619532391648883280063, −4.33772352813683367756699412549, −3.34956595105040784799329489917, −2.83079560782821657842389682170, −1.73367994208127801224848440072, −0.73255494986985676588407775757,
0.73255494986985676588407775757, 1.73367994208127801224848440072, 2.83079560782821657842389682170, 3.34956595105040784799329489917, 4.33772352813683367756699412549, 5.01986966619532391648883280063, 5.96454399954361730152367859326, 6.47747047049014084290698821884, 6.85420855449600597353886994838, 8.026407708054951337820521172458
