| L(s) = 1 | + 3.82·5-s + 7-s + 0.828·11-s − 2.82·13-s − 5.82·17-s + 1.17·19-s + 4·23-s + 9.65·25-s + 2.82·29-s − 1.17·31-s + 3.82·35-s − 8.65·37-s + 1.82·41-s + 2.17·43-s + 2.65·47-s + 49-s + 2·53-s + 3.17·55-s + 10.6·59-s + 8.82·61-s − 10.8·65-s + 4·67-s + 9.65·71-s + 5.65·73-s + 0.828·77-s + 3.82·79-s + 16.6·83-s + ⋯ |
| L(s) = 1 | + 1.71·5-s + 0.377·7-s + 0.249·11-s − 0.784·13-s − 1.41·17-s + 0.268·19-s + 0.834·23-s + 1.93·25-s + 0.525·29-s − 0.210·31-s + 0.647·35-s − 1.42·37-s + 0.285·41-s + 0.331·43-s + 0.387·47-s + 0.142·49-s + 0.274·53-s + 0.427·55-s + 1.38·59-s + 1.13·61-s − 1.34·65-s + 0.488·67-s + 1.14·71-s + 0.662·73-s + 0.0944·77-s + 0.430·79-s + 1.82·83-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
| \(L(1)\) |
\(\approx\) |
\(2.968556848\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.968556848\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.82T + 5T^{2} \) |
| 11 | \( 1 - 0.828T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 5.82T + 17T^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 - 2.82T + 29T^{2} \) |
| 31 | \( 1 + 1.17T + 31T^{2} \) |
| 37 | \( 1 + 8.65T + 37T^{2} \) |
| 41 | \( 1 - 1.82T + 41T^{2} \) |
| 43 | \( 1 - 2.17T + 43T^{2} \) |
| 47 | \( 1 - 2.65T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 - 8.82T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 - 9.65T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 - 3.82T + 79T^{2} \) |
| 83 | \( 1 - 16.6T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 10.1T + 97T^{2} \) |
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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
−8.212996953299786819172158904418, −7.04398942623686632923681948189, −6.76280820876178801777145553692, −5.91041776744424273735103968202, −5.15677400553642259383004422690, −4.75544454135554501554392066937, −3.58617760969674304067482973091, −2.38106484880499421437002644087, −2.10066745452234160485159281233, −0.921502306469921803809596748706,
0.921502306469921803809596748706, 2.10066745452234160485159281233, 2.38106484880499421437002644087, 3.58617760969674304067482973091, 4.75544454135554501554392066937, 5.15677400553642259383004422690, 5.91041776744424273735103968202, 6.76280820876178801777145553692, 7.04398942623686632923681948189, 8.212996953299786819172158904418
