| L(s) = 1 | − 4.76·3-s − 2.23i·5-s − 7.05i·7-s + 13.6·9-s − 10.4i·11-s + 19.0·13-s + 10.6i·15-s + 12.8i·17-s + 22.7i·19-s + 33.6i·21-s + (6.14 + 22.1i)23-s − 5.00·25-s − 22.3·27-s − 8.61·29-s − 22.2·31-s + ⋯ |
| L(s) = 1 | − 1.58·3-s − 0.447i·5-s − 1.00i·7-s + 1.52·9-s − 0.950i·11-s + 1.46·13-s + 0.710i·15-s + 0.754i·17-s + 1.19i·19-s + 1.60i·21-s + (0.266 + 0.963i)23-s − 0.200·25-s − 0.827·27-s − 0.297·29-s − 0.716·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.963 - 0.266i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9657411032\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9657411032\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 23 | \( 1 + (-6.14 - 22.1i)T \) |
| good | 3 | \( 1 + 4.76T + 9T^{2} \) |
| 7 | \( 1 + 7.05iT - 49T^{2} \) |
| 11 | \( 1 + 10.4iT - 121T^{2} \) |
| 13 | \( 1 - 19.0T + 169T^{2} \) |
| 17 | \( 1 - 12.8iT - 289T^{2} \) |
| 19 | \( 1 - 22.7iT - 361T^{2} \) |
| 29 | \( 1 + 8.61T + 841T^{2} \) |
| 31 | \( 1 + 22.2T + 961T^{2} \) |
| 37 | \( 1 + 29.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 18.7T + 1.68e3T^{2} \) |
| 43 | \( 1 + 10.0iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 20.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 17.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 103.T + 3.48e3T^{2} \) |
| 61 | \( 1 - 74.6iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 101. iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 69.4T + 5.04e3T^{2} \) |
| 73 | \( 1 + 122.T + 5.32e3T^{2} \) |
| 79 | \( 1 + 140. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 126. iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 11.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 57.0iT - 9.40e3T^{2} \) |
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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
−9.092630839099466052441618035531, −8.274338931811164353363788950316, −7.39940315698499766226291277726, −6.48971795350341377641982517591, −5.77276190329332492412232868470, −5.38215518331232548614452106051, −4.01710919535969639142822494377, −3.69309520634639895651473076909, −1.46755162026363560439918106665, −0.823381990938221509340697357126,
0.45985987390455745902790483605, 1.83039672072311190754237782987, 3.02964790786379054162826837816, 4.37217737800119481286849499499, 5.12635264210614507738490135597, 5.79659043240278292207039959107, 6.65779022258009542234215205486, 6.99204955605657902471984243371, 8.292256227025200224632548263417, 9.133508460420484442936430655828
