| L(s) = 1 | − 3-s + 3.46·5-s − 7-s + 9-s − 11-s + 2·13-s − 3.46·15-s − 3.46·17-s + 1.46·19-s + 21-s + 6.92·23-s + 6.99·25-s − 27-s − 6·29-s + 1.46·31-s + 33-s − 3.46·35-s + 8.92·37-s − 2·39-s − 3.46·41-s − 2.92·43-s + 3.46·45-s − 2.53·47-s + 49-s + 3.46·51-s + 12.9·53-s − 3.46·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.54·5-s − 0.377·7-s + 0.333·9-s − 0.301·11-s + 0.554·13-s − 0.894·15-s − 0.840·17-s + 0.335·19-s + 0.218·21-s + 1.44·23-s + 1.39·25-s − 0.192·27-s − 1.11·29-s + 0.262·31-s + 0.174·33-s − 0.585·35-s + 1.46·37-s − 0.320·39-s − 0.541·41-s − 0.446·43-s + 0.516·45-s − 0.369·47-s + 0.142·49-s + 0.485·51-s + 1.77·53-s − 0.467·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.058353549\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.058353549\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3.46T + 5T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 - 1.46T + 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 - 8.92T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 2.92T + 43T^{2} \) |
| 47 | \( 1 + 2.53T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 6.92T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 1.07T + 67T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + 7.46T + 73T^{2} \) |
| 79 | \( 1 + 2.92T + 79T^{2} \) |
| 83 | \( 1 - 16.3T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.805962189400321519760481662774, −7.67680656666996829959654299417, −6.71718612806263179880638356118, −6.36022152201300029403181710169, −5.48218312997082777041628010177, −5.07886178544468320944123398987, −3.93687074178094711501631682351, −2.81271216966431650299620767270, −1.97203411663048197608534687683, −0.883412126452799354150545951932,
0.883412126452799354150545951932, 1.97203411663048197608534687683, 2.81271216966431650299620767270, 3.93687074178094711501631682351, 5.07886178544468320944123398987, 5.48218312997082777041628010177, 6.36022152201300029403181710169, 6.71718612806263179880638356118, 7.67680656666996829959654299417, 8.805962189400321519760481662774
