| L(s) = 1 | + (−1.15 − 1.29i)3-s − 5-s − 4.31i·7-s + (−0.329 + 2.98i)9-s − 5.96i·11-s − 4.89i·13-s + (1.15 + 1.29i)15-s − 1.07i·17-s + 4.23·19-s + (−5.56 + 4.98i)21-s + 5.38·23-s + 25-s + (4.22 − 3.02i)27-s + 2.80·29-s − 0.927i·31-s + ⋯ |
| L(s) = 1 | + (−0.667 − 0.744i)3-s − 0.447·5-s − 1.62i·7-s + (−0.109 + 0.993i)9-s − 1.79i·11-s − 1.35i·13-s + (0.298 + 0.333i)15-s − 0.260i·17-s + 0.971·19-s + (−1.21 + 1.08i)21-s + 1.12·23-s + 0.200·25-s + (0.813 − 0.581i)27-s + 0.520·29-s − 0.166i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0549i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.107709648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107709648\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.15 + 1.29i)T \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 + 4.31iT - 7T^{2} \) |
| 11 | \( 1 + 5.96iT - 11T^{2} \) |
| 13 | \( 1 + 4.89iT - 13T^{2} \) |
| 17 | \( 1 + 1.07iT - 17T^{2} \) |
| 19 | \( 1 - 4.23T + 19T^{2} \) |
| 23 | \( 1 - 5.38T + 23T^{2} \) |
| 29 | \( 1 - 2.80T + 29T^{2} \) |
| 31 | \( 1 + 0.927iT - 31T^{2} \) |
| 37 | \( 1 + 8.35iT - 37T^{2} \) |
| 41 | \( 1 - 6.50iT - 41T^{2} \) |
| 43 | \( 1 + 11.4T + 43T^{2} \) |
| 47 | \( 1 - 1.92T + 47T^{2} \) |
| 53 | \( 1 - 1.46T + 53T^{2} \) |
| 59 | \( 1 - 5.34iT - 59T^{2} \) |
| 61 | \( 1 + 5.42iT - 61T^{2} \) |
| 67 | \( 1 - 3.47T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 - 14.4T + 73T^{2} \) |
| 79 | \( 1 + 1.91iT - 79T^{2} \) |
| 83 | \( 1 - 3.20iT - 83T^{2} \) |
| 89 | \( 1 - 0.538iT - 89T^{2} \) |
| 97 | \( 1 + 7.78T + 97T^{2} \) |
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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
−8.438444563164458425203171451351, −7.920237560113347185591863434972, −7.25152617497526954317850363476, −6.55291367651525928956868394095, −5.59025784724038575562810060172, −4.91258928838860113185593137370, −3.64090648598229838471637178930, −2.97981069460164402230497758233, −1.00637504819323588837761882412, −0.57169716556450281076206799637,
1.68647726734249860658071370337, 2.84953620721357525904735571481, 3.99655578847760957284909093443, 4.95206822273654236458614560625, 5.24584090681045802844229475883, 6.56097965965374842478493857947, 6.93312929107826766886767079999, 8.206490958502265327343273037999, 9.093723899139857710107787048037, 9.491522724393663471160387128067
