| L(s) = 1 | − 0.732·5-s + 7-s + 4·11-s − 1.46·13-s + 3.26·17-s + 19-s + 6.92·23-s − 4.46·25-s + 3.26·29-s − 2·31-s − 0.732·35-s − 4.92·37-s + 8.92·41-s + 4.92·43-s − 0.196·47-s + 49-s − 7.66·53-s − 2.92·55-s + 10.9·59-s + 0.928·61-s + 1.07·65-s + 5.46·67-s − 4.19·71-s − 10.3·73-s + 4·77-s + 2.92·79-s − 8.19·83-s + ⋯ |
| L(s) = 1 | − 0.327·5-s + 0.377·7-s + 1.20·11-s − 0.406·13-s + 0.792·17-s + 0.229·19-s + 1.44·23-s − 0.892·25-s + 0.606·29-s − 0.359·31-s − 0.123·35-s − 0.810·37-s + 1.39·41-s + 0.751·43-s − 0.0286·47-s + 0.142·49-s − 1.05·53-s − 0.394·55-s + 1.42·59-s + 0.118·61-s + 0.132·65-s + 0.667·67-s − 0.497·71-s − 1.21·73-s + 0.455·77-s + 0.329·79-s − 0.899·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.363916017\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.363916017\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + 0.732T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 - 3.26T + 17T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 - 3.26T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 + 4.92T + 37T^{2} \) |
| 41 | \( 1 - 8.92T + 41T^{2} \) |
| 43 | \( 1 - 4.92T + 43T^{2} \) |
| 47 | \( 1 + 0.196T + 47T^{2} \) |
| 53 | \( 1 + 7.66T + 53T^{2} \) |
| 59 | \( 1 - 10.9T + 59T^{2} \) |
| 61 | \( 1 - 0.928T + 61T^{2} \) |
| 67 | \( 1 - 5.46T + 67T^{2} \) |
| 71 | \( 1 + 4.19T + 71T^{2} \) |
| 73 | \( 1 + 10.3T + 73T^{2} \) |
| 79 | \( 1 - 2.92T + 79T^{2} \) |
| 83 | \( 1 + 8.19T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 11.8T + 97T^{2} \) |
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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
−7.56567840244980236279477988784, −7.14757172655553317620861811931, −6.36147310357871424783045138282, −5.61475450422330316634752626865, −4.91240642106074471171370359062, −4.15824968975389268927721359387, −3.51292159455694548582057254415, −2.66624203473833713799894779997, −1.59512035975531476826909318526, −0.78338574392585962006356418634,
0.78338574392585962006356418634, 1.59512035975531476826909318526, 2.66624203473833713799894779997, 3.51292159455694548582057254415, 4.15824968975389268927721359387, 4.91240642106074471171370359062, 5.61475450422330316634752626865, 6.36147310357871424783045138282, 7.14757172655553317620861811931, 7.56567840244980236279477988784
