| L(s) = 1 | + 2.56·5-s − 7-s + 5.12·11-s − 2.43·13-s + 4.12·17-s + 3.12·19-s + 8.68·23-s + 1.56·25-s + 9.56·29-s + 1.56·31-s − 2.56·35-s − 0.561·37-s − 7.43·41-s − 7.24·43-s + 0.561·47-s + 49-s − 6.68·53-s + 13.1·55-s − 8.12·59-s − 8.24·61-s − 6.24·65-s + 3.31·67-s + 1.56·71-s − 7.12·73-s − 5.12·77-s + 8.56·79-s + 2.56·83-s + ⋯ |
| L(s) = 1 | + 1.14·5-s − 0.377·7-s + 1.54·11-s − 0.676·13-s + 0.999·17-s + 0.716·19-s + 1.81·23-s + 0.312·25-s + 1.77·29-s + 0.280·31-s − 0.432·35-s − 0.0923·37-s − 1.16·41-s − 1.10·43-s + 0.0819·47-s + 0.142·49-s − 0.918·53-s + 1.76·55-s − 1.05·59-s − 1.05·61-s − 0.774·65-s + 0.405·67-s + 0.185·71-s − 0.833·73-s − 0.583·77-s + 0.963·79-s + 0.281·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.982885032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.982885032\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| good | 5 | \( 1 - 2.56T + 5T^{2} \) |
| 11 | \( 1 - 5.12T + 11T^{2} \) |
| 13 | \( 1 + 2.43T + 13T^{2} \) |
| 17 | \( 1 - 4.12T + 17T^{2} \) |
| 19 | \( 1 - 3.12T + 19T^{2} \) |
| 23 | \( 1 - 8.68T + 23T^{2} \) |
| 29 | \( 1 - 9.56T + 29T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 + 0.561T + 37T^{2} \) |
| 41 | \( 1 + 7.43T + 41T^{2} \) |
| 43 | \( 1 + 7.24T + 43T^{2} \) |
| 47 | \( 1 - 0.561T + 47T^{2} \) |
| 53 | \( 1 + 6.68T + 53T^{2} \) |
| 59 | \( 1 + 8.12T + 59T^{2} \) |
| 61 | \( 1 + 8.24T + 61T^{2} \) |
| 67 | \( 1 - 3.31T + 67T^{2} \) |
| 71 | \( 1 - 1.56T + 71T^{2} \) |
| 73 | \( 1 + 7.12T + 73T^{2} \) |
| 79 | \( 1 - 8.56T + 79T^{2} \) |
| 83 | \( 1 - 2.56T + 83T^{2} \) |
| 89 | \( 1 + 1.31T + 89T^{2} \) |
| 97 | \( 1 + 1.75T + 97T^{2} \) |
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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
−8.116449456765153032038800563412, −7.15250902348865712865562849079, −6.60745435867943226993054356340, −6.07471152038805856006886508985, −5.16660773076303965196930016671, −4.65241313871062041872323115592, −3.39174930978679927647151231476, −2.91645301135024965335950802141, −1.69378607093471906529247316726, −0.995088022773706210693827393999,
0.995088022773706210693827393999, 1.69378607093471906529247316726, 2.91645301135024965335950802141, 3.39174930978679927647151231476, 4.65241313871062041872323115592, 5.16660773076303965196930016671, 6.07471152038805856006886508985, 6.60745435867943226993054356340, 7.15250902348865712865562849079, 8.116449456765153032038800563412
