| L(s) = 1 | − 2.23i·5-s + 0.509i·7-s + 17.4i·11-s + 16.2·13-s − 10.8i·17-s + 9.45i·19-s + (0.700 − 22.9i)23-s − 5.00·25-s − 20.6·29-s + 11.5·31-s + 1.14·35-s − 62.7i·37-s − 12.0·41-s − 37.7i·43-s + 83.3·47-s + ⋯ |
| L(s) = 1 | − 0.447i·5-s + 0.0728i·7-s + 1.58i·11-s + 1.25·13-s − 0.638i·17-s + 0.497i·19-s + (0.0304 − 0.999i)23-s − 0.200·25-s − 0.713·29-s + 0.370·31-s + 0.0325·35-s − 1.69i·37-s − 0.294·41-s − 0.878i·43-s + 1.77·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.266502994\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.266502994\) |
| \(L(2)\) |
|
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 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-0.700 + 22.9i)T \) |
| good | 7 | \( 1 - 0.509iT - 49T^{2} \) |
| 11 | \( 1 - 17.4iT - 121T^{2} \) |
| 13 | \( 1 - 16.2T + 169T^{2} \) |
| 17 | \( 1 + 10.8iT - 289T^{2} \) |
| 19 | \( 1 - 9.45iT - 361T^{2} \) |
| 29 | \( 1 + 20.6T + 841T^{2} \) |
| 31 | \( 1 - 11.5T + 961T^{2} \) |
| 37 | \( 1 + 62.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 12.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 37.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 83.3T + 2.20e3T^{2} \) |
| 53 | \( 1 - 58.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 50.0T + 3.48e3T^{2} \) |
| 61 | \( 1 + 38.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 82.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 125.T + 5.04e3T^{2} \) |
| 73 | \( 1 - 138.T + 5.32e3T^{2} \) |
| 79 | \( 1 - 111. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 40.8iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 45.4iT - 9.40e3T^{2} \) |
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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
−8.278421756242599377533740645894, −7.44699688452194935005776889237, −6.91931928194866827068695669719, −5.93461857842025192444972188802, −5.32547521509916368420615335833, −4.32032081871796290630894882063, −3.91027390550045429127660983985, −2.59066247131000122318855565222, −1.78329207343034940176872102162, −0.70443469846338148996688943983,
0.71311684244378576263828443540, 1.67474779387358332437361417089, 3.02242947100608775419185792265, 3.47144522000163652056586274720, 4.34306124044296838949286013586, 5.53547768543703052118592549488, 6.02672245164412258249970654444, 6.65680123878957618718235356817, 7.59878263250441063839541135532, 8.330784098989466128851177719299
