| L(s) = 1 | − 1.81i·3-s − 4.91i·7-s − 0.289·9-s + 0.578·11-s + 6.39i·13-s − 0.710i·17-s − 19-s − 8.91·21-s + 2.71i·23-s − 4.91i·27-s − 6.54·29-s + 1.42·31-s − 1.04i·33-s − 9.10i·37-s + 11.5·39-s + ⋯ |
| L(s) = 1 | − 1.04i·3-s − 1.85i·7-s − 0.0963·9-s + 0.174·11-s + 1.77i·13-s − 0.172i·17-s − 0.229·19-s − 1.94·21-s + 0.565i·23-s − 0.946i·27-s − 1.21·29-s + 0.255·31-s − 0.182i·33-s − 1.49i·37-s + 1.85·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 - 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9304687869\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9304687869\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 + 1.81iT - 3T^{2} \) |
| 7 | \( 1 + 4.91iT - 7T^{2} \) |
| 11 | \( 1 - 0.578T + 11T^{2} \) |
| 13 | \( 1 - 6.39iT - 13T^{2} \) |
| 17 | \( 1 + 0.710iT - 17T^{2} \) |
| 23 | \( 1 - 2.71iT - 23T^{2} \) |
| 29 | \( 1 + 6.54T + 29T^{2} \) |
| 31 | \( 1 - 1.42T + 31T^{2} \) |
| 37 | \( 1 + 9.10iT - 37T^{2} \) |
| 41 | \( 1 + 11.0T + 41T^{2} \) |
| 43 | \( 1 + 5.83iT - 43T^{2} \) |
| 47 | \( 1 + 1.15iT - 47T^{2} \) |
| 53 | \( 1 + 13.2iT - 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 9.04T + 61T^{2} \) |
| 67 | \( 1 - 2.97iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 9.38iT - 73T^{2} \) |
| 79 | \( 1 + 4.37T + 79T^{2} \) |
| 83 | \( 1 - 0.372iT - 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 3.94iT - 97T^{2} \) |
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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
−7.75564306029301909005937112793, −7.21291453605519047188875767917, −6.87373556747472887758562148297, −6.19219775938170273675569411027, −4.94023701616922279020820252619, −4.09251266842098676810205321403, −3.59863614099654085249728311458, −1.99254294041878720120639405390, −1.46254025385839610955371610446, −0.26018709799994795815547576798,
1.64797676790851794209185882959, 2.89248786738415900704407369059, 3.28791847174755175056626734007, 4.55508921922315405766739992150, 5.12491994060083212805786109009, 5.82071091203150962088547733840, 6.41515768520783005186014679241, 7.72862548573655027285975671615, 8.293149498226244128530018772562, 9.076733012085599844860043102850
