| L(s) = 1 | − 2.09·3-s + 3.70·5-s + 7-s + 1.40·9-s + 0.626·11-s + 0.872·13-s − 7.77·15-s + 7.62·19-s − 2.09·21-s + 1.64·23-s + 8.71·25-s + 3.34·27-s + 9.06·29-s − 1.14·31-s − 1.31·33-s + 3.70·35-s − 8.60·37-s − 1.83·39-s − 4.78·41-s + 3.44·43-s + 5.19·45-s + 9.39·47-s + 49-s − 13.2·53-s + 2.31·55-s − 15.9·57-s − 3.73·59-s + ⋯ |
| L(s) = 1 | − 1.21·3-s + 1.65·5-s + 0.377·7-s + 0.467·9-s + 0.188·11-s + 0.242·13-s − 2.00·15-s + 1.74·19-s − 0.457·21-s + 0.342·23-s + 1.74·25-s + 0.644·27-s + 1.68·29-s − 0.205·31-s − 0.228·33-s + 0.625·35-s − 1.41·37-s − 0.293·39-s − 0.747·41-s + 0.525·43-s + 0.775·45-s + 1.37·47-s + 0.142·49-s − 1.82·53-s + 0.312·55-s − 2.11·57-s − 0.486·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8092 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.258410150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.258410150\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 \) |
| good | 3 | \( 1 + 2.09T + 3T^{2} \) |
| 5 | \( 1 - 3.70T + 5T^{2} \) |
| 11 | \( 1 - 0.626T + 11T^{2} \) |
| 13 | \( 1 - 0.872T + 13T^{2} \) |
| 19 | \( 1 - 7.62T + 19T^{2} \) |
| 23 | \( 1 - 1.64T + 23T^{2} \) |
| 29 | \( 1 - 9.06T + 29T^{2} \) |
| 31 | \( 1 + 1.14T + 31T^{2} \) |
| 37 | \( 1 + 8.60T + 37T^{2} \) |
| 41 | \( 1 + 4.78T + 41T^{2} \) |
| 43 | \( 1 - 3.44T + 43T^{2} \) |
| 47 | \( 1 - 9.39T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 + 3.73T + 59T^{2} \) |
| 61 | \( 1 - 0.372T + 61T^{2} \) |
| 67 | \( 1 - 5.41T + 67T^{2} \) |
| 71 | \( 1 - 6.58T + 71T^{2} \) |
| 73 | \( 1 - 6.87T + 73T^{2} \) |
| 79 | \( 1 - 5.14T + 79T^{2} \) |
| 83 | \( 1 - 0.685T + 83T^{2} \) |
| 89 | \( 1 - 4.68T + 89T^{2} \) |
| 97 | \( 1 - 15.5T + 97T^{2} \) |
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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
−7.69989736909254009786520258024, −6.76758386341777668827683056856, −6.40542845363873741208510673123, −5.65395489005520874108310520107, −5.19294421486560295165010532883, −4.75988476437987802872986071676, −3.42731546509180598342122864323, −2.55351764000380629095274759064, −1.52062208797326917665208480126, −0.872541074920551166068366314193,
0.872541074920551166068366314193, 1.52062208797326917665208480126, 2.55351764000380629095274759064, 3.42731546509180598342122864323, 4.75988476437987802872986071676, 5.19294421486560295165010532883, 5.65395489005520874108310520107, 6.40542845363873741208510673123, 6.76758386341777668827683056856, 7.69989736909254009786520258024
