| L(s) = 1 | + (1.13 − 2.77i)3-s + 5.85i·7-s + (−6.42 − 6.29i)9-s − 12.4i·11-s − 11.8i·13-s + 29.2·17-s + 3.19·19-s + (16.2 + 6.64i)21-s − 19.5·23-s + (−24.7 + 10.7i)27-s + 30.3i·29-s − 3.57·31-s + (−34.6 − 14.1i)33-s − 42.7i·37-s + (−32.9 − 13.4i)39-s + ⋯ |
| L(s) = 1 | + (0.377 − 0.925i)3-s + 0.836i·7-s + (−0.714 − 0.699i)9-s − 1.13i·11-s − 0.912i·13-s + 1.72·17-s + 0.168·19-s + (0.774 + 0.316i)21-s − 0.849·23-s + (−0.917 + 0.396i)27-s + 1.04i·29-s − 0.115·31-s + (−1.04 − 0.428i)33-s − 1.15i·37-s + (−0.844 − 0.344i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.752 + 0.659i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.752 + 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.663068399\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.663068399\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.13 + 2.77i)T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 5.85iT - 49T^{2} \) |
| 11 | \( 1 + 12.4iT - 121T^{2} \) |
| 13 | \( 1 + 11.8iT - 169T^{2} \) |
| 17 | \( 1 - 29.2T + 289T^{2} \) |
| 19 | \( 1 - 3.19T + 361T^{2} \) |
| 23 | \( 1 + 19.5T + 529T^{2} \) |
| 29 | \( 1 - 30.3iT - 841T^{2} \) |
| 31 | \( 1 + 3.57T + 961T^{2} \) |
| 37 | \( 1 + 42.7iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 6.39iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 62.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 69.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 57.7T + 2.80e3T^{2} \) |
| 59 | \( 1 + 78.9iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 68.5T + 3.72e3T^{2} \) |
| 67 | \( 1 + 90.5iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 26.5iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 40.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 148.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 9.36T + 6.88e3T^{2} \) |
| 89 | \( 1 - 109. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 161. iT - 9.40e3T^{2} \) |
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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
−9.057256819941717562569231392153, −8.245577382128197338037025797944, −7.84152757872463743878652859335, −6.77486223123983746423468185048, −5.69497139789948904175328343659, −5.48066328062898512024832118844, −3.52562085234804882460432539299, −2.97833491948852718762693253433, −1.71337650613985413659556444419, −0.46760084898397587065397776435,
1.49490905993385134060603054116, 2.83415938534910400794249922916, 3.94768914215378202305349871398, 4.48548617973235588580654044723, 5.47449825557458900071857174258, 6.59610632089509880742296561009, 7.62340865227121799864664507953, 8.134215810105915549499470685833, 9.337982135730288755773477571745, 9.980276295265376131521469888240
