| L(s) = 1 | + 0.268·2-s + 1.14·3-s − 1.92·4-s − 2.56·5-s + 0.306·6-s − 1.05·8-s − 1.69·9-s − 0.686·10-s − 3.94·11-s − 2.20·12-s − 2.92·15-s + 3.57·16-s − 0.785·17-s − 0.454·18-s − 7.49·19-s + 4.93·20-s − 1.05·22-s − 7.95·23-s − 1.20·24-s + 1.56·25-s − 5.36·27-s + 2.35·29-s − 0.785·30-s − 2.55·31-s + 3.06·32-s − 4.51·33-s − 0.210·34-s + ⋯ |
| L(s) = 1 | + 0.189·2-s + 0.659·3-s − 0.964·4-s − 1.14·5-s + 0.125·6-s − 0.372·8-s − 0.564·9-s − 0.217·10-s − 1.18·11-s − 0.636·12-s − 0.756·15-s + 0.893·16-s − 0.190·17-s − 0.107·18-s − 1.71·19-s + 1.10·20-s − 0.225·22-s − 1.65·23-s − 0.245·24-s + 0.312·25-s − 1.03·27-s + 0.436·29-s − 0.143·30-s − 0.458·31-s + 0.541·32-s − 0.785·33-s − 0.0361·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06138602203\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06138602203\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 0.268T + 2T^{2} \) |
| 3 | \( 1 - 1.14T + 3T^{2} \) |
| 5 | \( 1 + 2.56T + 5T^{2} \) |
| 11 | \( 1 + 3.94T + 11T^{2} \) |
| 17 | \( 1 + 0.785T + 17T^{2} \) |
| 19 | \( 1 + 7.49T + 19T^{2} \) |
| 23 | \( 1 + 7.95T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 + 2.55T + 31T^{2} \) |
| 37 | \( 1 + 6.75T + 37T^{2} \) |
| 41 | \( 1 + 2.43T + 41T^{2} \) |
| 43 | \( 1 + 2.24T + 43T^{2} \) |
| 47 | \( 1 - 1.31T + 47T^{2} \) |
| 53 | \( 1 - 9.27T + 53T^{2} \) |
| 59 | \( 1 + 8.96T + 59T^{2} \) |
| 61 | \( 1 + 9.44T + 61T^{2} \) |
| 67 | \( 1 - 1.35T + 67T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 - 0.768T + 73T^{2} \) |
| 79 | \( 1 - 6.19T + 79T^{2} \) |
| 83 | \( 1 + 1.07T + 83T^{2} \) |
| 89 | \( 1 - 7.66T + 89T^{2} \) |
| 97 | \( 1 + 2.37T + 97T^{2} \) |
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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
−7.937742183851696617987276263876, −7.47978660546454175975025100157, −6.33198272601142044055523217306, −5.64438026605902800878176790042, −4.83337810533743169646241430669, −4.14013822602896941263247732612, −3.64491283110275752117144736944, −2.83903014321783801968820330978, −1.96261037083161603607083263733, −0.10710580178647573481117557401,
0.10710580178647573481117557401, 1.96261037083161603607083263733, 2.83903014321783801968820330978, 3.64491283110275752117144736944, 4.14013822602896941263247732612, 4.83337810533743169646241430669, 5.64438026605902800878176790042, 6.33198272601142044055523217306, 7.47978660546454175975025100157, 7.937742183851696617987276263876
