| L(s) = 1 | − 4.83i·3-s + 2.23·5-s − 3.72·7-s − 14.3·9-s − 13.8·11-s + 4.10i·13-s − 10.8i·15-s − 15.3·17-s + (−16.4 − 9.49i)19-s + 18.0i·21-s + 17.4·23-s + 5.00·25-s + 25.8i·27-s − 16.3i·29-s − 15.9i·31-s + ⋯ |
| L(s) = 1 | − 1.61i·3-s + 0.447·5-s − 0.532·7-s − 1.59·9-s − 1.26·11-s + 0.315i·13-s − 0.720i·15-s − 0.900·17-s + (−0.866 − 0.499i)19-s + 0.857i·21-s + 0.758·23-s + 0.200·25-s + 0.956i·27-s − 0.565i·29-s − 0.513i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 380 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.866 - 0.499i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.162854 + 0.608336i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.162854 + 0.608336i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 19 | \( 1 + (16.4 + 9.49i)T \) |
| good | 3 | \( 1 + 4.83iT - 9T^{2} \) |
| 7 | \( 1 + 3.72T + 49T^{2} \) |
| 11 | \( 1 + 13.8T + 121T^{2} \) |
| 13 | \( 1 - 4.10iT - 169T^{2} \) |
| 17 | \( 1 + 15.3T + 289T^{2} \) |
| 23 | \( 1 - 17.4T + 529T^{2} \) |
| 29 | \( 1 + 16.3iT - 841T^{2} \) |
| 31 | \( 1 + 15.9iT - 961T^{2} \) |
| 37 | \( 1 - 5.14iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 42.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 1.56T + 1.84e3T^{2} \) |
| 47 | \( 1 + 54.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 55.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 101. iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 80.5T + 3.72e3T^{2} \) |
| 67 | \( 1 - 24.8iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 121. iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 33.9T + 5.32e3T^{2} \) |
| 79 | \( 1 + 102. iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 80.3T + 6.88e3T^{2} \) |
| 89 | \( 1 - 12.0iT - 7.92e3T^{2} \) |
| 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
−10.78863810615168565504684866662, −9.607789871203475609182069729909, −8.535185421384409403049487515318, −7.69806724284759803154389178326, −6.70423324626515923235867280595, −6.15435875673608137608839963221, −4.83512574163005432885207236649, −2.85845964860520055916338030690, −1.94759574677802967939741074008, −0.25563191469079571421240129357,
2.56826010512200650257849219027, 3.67588901472847422706642198314, 4.85230406410982218349889284945, 5.56981188435402044662450061401, 6.80724710633807841515320861380, 8.323526560298518888870120980009, 9.086577099661986300405465464398, 10.04498058198207616280316313458, 10.51069383358980139402848649619, 11.23871443453769061457652779459
