| L(s) = 1 | + (2.22 + 0.166i)5-s + 1.74i·7-s − 2.39·11-s − 3.50i·13-s − 5.88i·17-s − 1.93·19-s − i·23-s + (4.94 + 0.742i)25-s − 4.22·29-s + 1.61·31-s + (−0.289 + 3.88i)35-s − 4.06i·37-s − 8.46·41-s − 2.70i·43-s − 8.18i·47-s + ⋯ |
| L(s) = 1 | + (0.997 + 0.0744i)5-s + 0.658i·7-s − 0.723·11-s − 0.971i·13-s − 1.42i·17-s − 0.443·19-s − 0.208i·23-s + (0.988 + 0.148i)25-s − 0.783·29-s + 0.289·31-s + (−0.0489 + 0.656i)35-s − 0.668i·37-s − 1.32·41-s − 0.412i·43-s − 1.19i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0744 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0744 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.492882669\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.492882669\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.22 - 0.166i)T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 - 1.74iT - 7T^{2} \) |
| 11 | \( 1 + 2.39T + 11T^{2} \) |
| 13 | \( 1 + 3.50iT - 13T^{2} \) |
| 17 | \( 1 + 5.88iT - 17T^{2} \) |
| 19 | \( 1 + 1.93T + 19T^{2} \) |
| 29 | \( 1 + 4.22T + 29T^{2} \) |
| 31 | \( 1 - 1.61T + 31T^{2} \) |
| 37 | \( 1 + 4.06iT - 37T^{2} \) |
| 41 | \( 1 + 8.46T + 41T^{2} \) |
| 43 | \( 1 + 2.70iT - 43T^{2} \) |
| 47 | \( 1 + 8.18iT - 47T^{2} \) |
| 53 | \( 1 + 2.78iT - 53T^{2} \) |
| 59 | \( 1 + 12.5T + 59T^{2} \) |
| 61 | \( 1 - 3.35T + 61T^{2} \) |
| 67 | \( 1 + 2.16iT - 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 4.98iT - 73T^{2} \) |
| 79 | \( 1 - 6.25T + 79T^{2} \) |
| 83 | \( 1 + 1.55iT - 83T^{2} \) |
| 89 | \( 1 - 7.14T + 89T^{2} \) |
| 97 | \( 1 + 5.93iT - 97T^{2} \) |
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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
−8.298970210134755417885376825865, −7.47031663189525212500247085339, −6.74612488017083763817712286345, −5.82693301589621020382377519050, −5.36326532455468482646856429876, −4.73547467906434837013373667657, −3.33760836986070268313581863113, −2.62755878032850688562692433496, −1.89250238408292160478574064400, −0.39577747384121919508772272938,
1.35071990250668445191354349158, 2.06388623763980102476828537952, 3.14134183189849782738637237475, 4.15910718496895511359680756850, 4.82136201904302144513720380899, 5.79528514616899723841336363280, 6.33435766651649341258342886610, 7.07519441483860612359583283317, 7.909284791373264866902796207536, 8.640057451465954878206402205750
