| L(s) = 1 | − 2·11-s − 13-s − 3·19-s + 9·31-s − 3·37-s − 2·41-s + 3·43-s + 6·47-s − 4·59-s + 2·61-s + 5·67-s + 14·71-s − 73-s − 9·79-s + 6·83-s + 4·89-s + 14·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 0.603·11-s − 0.277·13-s − 0.688·19-s + 1.61·31-s − 0.493·37-s − 0.312·41-s + 0.457·43-s + 0.875·47-s − 0.520·59-s + 0.256·61-s + 0.610·67-s + 1.66·71-s − 0.117·73-s − 1.01·79-s + 0.658·83-s + 0.423·89-s + 1.42·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.093458357\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.093458357\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 9 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 3 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + T + p T^{2} \) |
| 79 | \( 1 + 9 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−13.11894896776452, −12.68787751581889, −12.34416243901047, −11.69680760819553, −11.41118731402532, −10.67069993405651, −10.38307810256193, −9.980921856928523, −9.346606490105465, −8.900819678515336, −8.281814028278357, −8.022589611757012, −7.368294670683156, −6.908435236859053, −6.350333969247672, −5.871821079649759, −5.314300377830270, −4.656637039147326, −4.425989937016776, −3.567503784522288, −3.134682540031330, −2.317394054106229, −2.096784367377844, −1.065974293566668, −0.4590087644980795,
0.4590087644980795, 1.065974293566668, 2.096784367377844, 2.317394054106229, 3.134682540031330, 3.567503784522288, 4.425989937016776, 4.656637039147326, 5.314300377830270, 5.871821079649759, 6.350333969247672, 6.908435236859053, 7.368294670683156, 8.022589611757012, 8.281814028278357, 8.900819678515336, 9.346606490105465, 9.980921856928523, 10.38307810256193, 10.67069993405651, 11.41118731402532, 11.69680760819553, 12.34416243901047, 12.68787751581889, 13.11894896776452
