L(s) = 1 | + 0.568i·3-s + (−2.03 − 0.918i)5-s + (1.76 + 1.96i)7-s + 2.67·9-s + 0.417i·11-s − 2.13·13-s + (0.521 − 1.15i)15-s − 0.314·17-s + 2.19·19-s + (−1.11 + 1.00i)21-s + 0.992·23-s + (3.31 + 3.74i)25-s + 3.22i·27-s − 4.35·29-s + 4.22·31-s + ⋯ |
L(s) = 1 | + 0.328i·3-s + (−0.911 − 0.410i)5-s + (0.667 + 0.744i)7-s + 0.892·9-s + 0.125i·11-s − 0.592·13-s + (0.134 − 0.299i)15-s − 0.0761·17-s + 0.503·19-s + (−0.244 + 0.219i)21-s + 0.206·23-s + (0.662 + 0.748i)25-s + 0.620i·27-s − 0.808·29-s + 0.759·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.459 - 0.888i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.456782830\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.456782830\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (2.03 + 0.918i)T \) |
| 7 | \( 1 + (-1.76 - 1.96i)T \) |
good | 3 | \( 1 - 0.568iT - 3T^{2} \) |
| 11 | \( 1 - 0.417iT - 11T^{2} \) |
| 13 | \( 1 + 2.13T + 13T^{2} \) |
| 17 | \( 1 + 0.314T + 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 23 | \( 1 - 0.992T + 23T^{2} \) |
| 29 | \( 1 + 4.35T + 29T^{2} \) |
| 31 | \( 1 - 4.22T + 31T^{2} \) |
| 37 | \( 1 - 6.83iT - 37T^{2} \) |
| 41 | \( 1 - 3.82iT - 41T^{2} \) |
| 43 | \( 1 - 6.04T + 43T^{2} \) |
| 47 | \( 1 - 4.98iT - 47T^{2} \) |
| 53 | \( 1 - 7.14iT - 53T^{2} \) |
| 59 | \( 1 - 9.50T + 59T^{2} \) |
| 61 | \( 1 - 0.783iT - 61T^{2} \) |
| 67 | \( 1 - 11.1T + 67T^{2} \) |
| 71 | \( 1 + 6.55iT - 71T^{2} \) |
| 73 | \( 1 + 5.11T + 73T^{2} \) |
| 79 | \( 1 - 11.4iT - 79T^{2} \) |
| 83 | \( 1 - 5.49iT - 83T^{2} \) |
| 89 | \( 1 - 12.5iT - 89T^{2} \) |
| 97 | \( 1 + 0.314T + 97T^{2} \) |
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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
−9.819540786688130801386168031462, −9.214082404712224164862112101459, −8.256631021874328099498651185636, −7.64466610794831972471395643111, −6.79380224223745444317398231434, −5.45361374800724395025299888831, −4.73058527191388933838868372984, −4.02028191799351808598893170161, −2.74139130965575516362088899179, −1.29688594769416943550985393627,
0.74423171699384352395929271917, 2.18681765600360825936154105737, 3.61978071473809381467413780736, 4.32069336907227078680300333651, 5.27585629559543030604551560391, 6.66722011220215820701190903284, 7.35537860099759099589748658424, 7.73728334198444465233369188470, 8.718449489337486683369177013314, 9.858894030035603694280906599715