| L(s) = 1 | − 3-s + 5-s + 4.30·7-s + 9-s + 3.72·11-s + 1.41·13-s − 15-s + 7.66·17-s + 6.88·19-s − 4.30·21-s + 1.22·23-s + 25-s − 27-s + 1.92·29-s − 31-s − 3.72·33-s + 4.30·35-s − 0.580·37-s − 1.41·39-s − 6.16·41-s − 10.0·43-s + 45-s + 6.82·47-s + 11.5·49-s − 7.66·51-s − 2.93·53-s + 3.72·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.62·7-s + 0.333·9-s + 1.12·11-s + 0.393·13-s − 0.258·15-s + 1.85·17-s + 1.58·19-s − 0.939·21-s + 0.255·23-s + 0.200·25-s − 0.192·27-s + 0.356·29-s − 0.179·31-s − 0.648·33-s + 0.728·35-s − 0.0954·37-s − 0.227·39-s − 0.962·41-s − 1.53·43-s + 0.149·45-s + 0.995·47-s + 1.65·49-s − 1.07·51-s − 0.403·53-s + 0.502·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.704469387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.704469387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 - 4.30T + 7T^{2} \) |
| 11 | \( 1 - 3.72T + 11T^{2} \) |
| 13 | \( 1 - 1.41T + 13T^{2} \) |
| 17 | \( 1 - 7.66T + 17T^{2} \) |
| 19 | \( 1 - 6.88T + 19T^{2} \) |
| 23 | \( 1 - 1.22T + 23T^{2} \) |
| 29 | \( 1 - 1.92T + 29T^{2} \) |
| 37 | \( 1 + 0.580T + 37T^{2} \) |
| 41 | \( 1 + 6.16T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 6.82T + 47T^{2} \) |
| 53 | \( 1 + 2.93T + 53T^{2} \) |
| 59 | \( 1 + 5.08T + 59T^{2} \) |
| 61 | \( 1 - 4.32T + 61T^{2} \) |
| 67 | \( 1 + 14.3T + 67T^{2} \) |
| 71 | \( 1 + 6.25T + 71T^{2} \) |
| 73 | \( 1 + 4.75T + 73T^{2} \) |
| 79 | \( 1 + 6.67T + 79T^{2} \) |
| 83 | \( 1 - 3.21T + 83T^{2} \) |
| 89 | \( 1 - 3.09T + 89T^{2} \) |
| 97 | \( 1 + 3.71T + 97T^{2} \) |
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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
−8.474148845033081173246404432270, −7.72084979787942254009373901934, −7.12756368347572001524938318639, −6.15552664081596617481868568397, −5.39069266793999639002870459164, −4.99333545545412449920876010351, −3.97626889780858741850904330414, −3.05716937084526963900372927564, −1.41533591809715689191173544359, −1.30800025792417976325922498544,
1.30800025792417976325922498544, 1.41533591809715689191173544359, 3.05716937084526963900372927564, 3.97626889780858741850904330414, 4.99333545545412449920876010351, 5.39069266793999639002870459164, 6.15552664081596617481868568397, 7.12756368347572001524938318639, 7.72084979787942254009373901934, 8.474148845033081173246404432270
