| L(s) = 1 | − 1.56i·2-s − 2.56i·3-s − 0.438·4-s − 4·6-s + i·7-s − 2.43i·8-s − 3.56·9-s + 2.56·11-s + 1.12i·12-s + 4.56i·13-s + 1.56·14-s − 4.68·16-s + 4.56i·17-s + 5.56i·18-s − 1.12·19-s + ⋯ |
| L(s) = 1 | − 1.10i·2-s − 1.47i·3-s − 0.219·4-s − 1.63·6-s + 0.377i·7-s − 0.862i·8-s − 1.18·9-s + 0.772·11-s + 0.324i·12-s + 1.26i·13-s + 0.417·14-s − 1.17·16-s + 1.10i·17-s + 1.31i·18-s − 0.257·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.286526 - 1.21374i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.286526 - 1.21374i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 2 | \( 1 + 1.56iT - 2T^{2} \) |
| 3 | \( 1 + 2.56iT - 3T^{2} \) |
| 11 | \( 1 - 2.56T + 11T^{2} \) |
| 13 | \( 1 - 4.56iT - 13T^{2} \) |
| 17 | \( 1 - 4.56iT - 17T^{2} \) |
| 19 | \( 1 + 1.12T + 19T^{2} \) |
| 23 | \( 1 + 5.12iT - 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 3.12T + 41T^{2} \) |
| 43 | \( 1 - 9.12iT - 43T^{2} \) |
| 47 | \( 1 + 3.68iT - 47T^{2} \) |
| 53 | \( 1 - 3.12iT - 53T^{2} \) |
| 59 | \( 1 - 4T + 59T^{2} \) |
| 61 | \( 1 + 9.36T + 61T^{2} \) |
| 67 | \( 1 - 6.24iT - 67T^{2} \) |
| 71 | \( 1 - 8T + 71T^{2} \) |
| 73 | \( 1 - 4.24iT - 73T^{2} \) |
| 79 | \( 1 - 6.56T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 - 14.8iT - 97T^{2} \) |
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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
−12.24400601530167099139224874168, −11.64724094259282630536278771321, −10.62858072811078557107656407839, −9.308538067901808379879984163159, −8.263573243497107765732447990748, −6.81917683160626994725379570752, −6.32854984995160221273028087113, −4.12956679267171323152191454580, −2.44097078858064708710288877194, −1.41353180269501736525083447880,
3.25111233793821000485396046763, 4.68008526645575399287219944177, 5.55326853687470932258968942019, 6.83270223698443783784000966318, 7.998732513706251784273081401710, 9.073878245378057383776968434764, 10.04453078925486213399057363670, 10.95578476841481094173342666857, 11.92970054487346081260274298709, 13.61157728120276420931803950597
