L(s) = 1 | + 0.363i·3-s + (1.75 − 1.38i)5-s + i·7-s + 2.86·9-s − 5.14·11-s − 4.64i·13-s + (0.504 + 0.636i)15-s + 3.86i·17-s − 0.778·19-s − 0.363·21-s − 5.00i·23-s + (1.14 − 4.86i)25-s + 2.13i·27-s + 9.42·29-s + 4.72·31-s + ⋯ |
L(s) = 1 | + 0.209i·3-s + (0.783 − 0.621i)5-s + 0.377i·7-s + 0.955·9-s − 1.55·11-s − 1.28i·13-s + (0.130 + 0.164i)15-s + 0.938i·17-s − 0.178·19-s − 0.0792·21-s − 1.04i·23-s + (0.228 − 0.973i)25-s + 0.410i·27-s + 1.74·29-s + 0.848·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.951107526\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.951107526\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-1.75 + 1.38i)T \) |
| 7 | \( 1 - iT \) |
good | 3 | \( 1 - 0.363iT - 3T^{2} \) |
| 11 | \( 1 + 5.14T + 11T^{2} \) |
| 13 | \( 1 + 4.64iT - 13T^{2} \) |
| 17 | \( 1 - 3.86iT - 17T^{2} \) |
| 19 | \( 1 + 0.778T + 19T^{2} \) |
| 23 | \( 1 + 5.00iT - 23T^{2} \) |
| 29 | \( 1 - 9.42T + 29T^{2} \) |
| 31 | \( 1 - 4.72T + 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 + 1.00T + 41T^{2} \) |
| 43 | \( 1 + 7.00iT - 43T^{2} \) |
| 47 | \( 1 + 11.4iT - 47T^{2} \) |
| 53 | \( 1 - 7.55iT - 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 11.5T + 61T^{2} \) |
| 67 | \( 1 - 11.7iT - 67T^{2} \) |
| 71 | \( 1 - 2.72T + 71T^{2} \) |
| 73 | \( 1 + 5.00iT - 73T^{2} \) |
| 79 | \( 1 + 5.68T + 79T^{2} \) |
| 83 | \( 1 + 4.67iT - 83T^{2} \) |
| 89 | \( 1 - 2.82T + 89T^{2} \) |
| 97 | \( 1 + 1.58iT - 97T^{2} \) |
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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
−8.688934370175680591022522687970, −8.391374543691498667619725130347, −7.49197175266775679555092912510, −6.44790611513350791462381533965, −5.62725566367529557867879909410, −5.03689476391723634347127321189, −4.23762783412656626156825203807, −2.89715030398171804771310048878, −2.10350572550630194825112209802, −0.71964500885534532917531131402,
1.27035579803350239097332096092, 2.35222366684003505636232296208, 3.14617134104138740823008942077, 4.51098445734869472639692642442, 5.04341725020676663427336038811, 6.24847097143829099066361432424, 6.79542325224046538095816571157, 7.50003957166020940662499917226, 8.216641807376576950122544900190, 9.433406174507939416767765535181