| L(s) = 1 | + (−1.14 − 1.30i)3-s + (−0.569 + 3.96i)5-s + (0.363 + 3.81i)7-s + (−0.390 + 2.97i)9-s + (4.23 − 1.69i)11-s + (−3.77 − 3.95i)13-s + (5.80 − 3.78i)15-s + (−6.20 + 2.14i)17-s + (−6.82 − 0.652i)19-s + (4.54 − 4.82i)21-s + (−0.473 + 0.0225i)23-s + (−10.5 − 3.10i)25-s + (4.31 − 2.89i)27-s + (7.56 − 4.36i)29-s + (−0.649 + 0.680i)31-s + ⋯ |
| L(s) = 1 | + (−0.659 − 0.751i)3-s + (−0.254 + 1.77i)5-s + (0.137 + 1.44i)7-s + (−0.130 + 0.991i)9-s + (1.27 − 0.511i)11-s + (−1.04 − 1.09i)13-s + (1.50 − 0.977i)15-s + (−1.50 + 0.520i)17-s + (−1.56 − 0.149i)19-s + (0.991 − 1.05i)21-s + (−0.0988 + 0.00470i)23-s + (−2.11 − 0.621i)25-s + (0.831 − 0.556i)27-s + (1.40 − 0.810i)29-s + (−0.116 + 0.122i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 804 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0652480 + 0.471533i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0652480 + 0.471533i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.14 + 1.30i)T \) |
| 67 | \( 1 + (2.35 - 7.83i)T \) |
| good | 5 | \( 1 + (0.569 - 3.96i)T + (-4.79 - 1.40i)T^{2} \) |
| 7 | \( 1 + (-0.363 - 3.81i)T + (-6.87 + 1.32i)T^{2} \) |
| 11 | \( 1 + (-4.23 + 1.69i)T + (7.96 - 7.59i)T^{2} \) |
| 13 | \( 1 + (3.77 + 3.95i)T + (-0.618 + 12.9i)T^{2} \) |
| 17 | \( 1 + (6.20 - 2.14i)T + (13.3 - 10.5i)T^{2} \) |
| 19 | \( 1 + (6.82 + 0.652i)T + (18.6 + 3.59i)T^{2} \) |
| 23 | \( 1 + (0.473 - 0.0225i)T + (22.8 - 2.18i)T^{2} \) |
| 29 | \( 1 + (-7.56 + 4.36i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (0.649 - 0.680i)T + (-1.47 - 30.9i)T^{2} \) |
| 37 | \( 1 + (0.455 - 0.789i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (0.877 + 0.169i)T + (38.0 + 15.2i)T^{2} \) |
| 43 | \( 1 + (-0.287 - 0.249i)T + (6.11 + 42.5i)T^{2} \) |
| 47 | \( 1 + (1.32 - 2.56i)T + (-27.2 - 38.2i)T^{2} \) |
| 53 | \( 1 + (2.31 + 2.67i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (-2.25 - 7.66i)T + (-49.6 + 31.8i)T^{2} \) |
| 61 | \( 1 + (-0.666 + 1.66i)T + (-44.1 - 42.0i)T^{2} \) |
| 71 | \( 1 + (-0.304 - 0.105i)T + (55.8 + 43.8i)T^{2} \) |
| 73 | \( 1 + (3.59 + 1.43i)T + (52.8 + 50.3i)T^{2} \) |
| 79 | \( 1 + (-1.01 + 0.246i)T + (70.2 - 36.1i)T^{2} \) |
| 83 | \( 1 + (7.37 - 9.38i)T + (-19.5 - 80.6i)T^{2} \) |
| 89 | \( 1 + (1.27 - 1.98i)T + (-36.9 - 80.9i)T^{2} \) |
| 97 | \( 1 + (13.0 + 7.55i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.84148048181732421915772737463, −10.07919255921195269141132761923, −8.734853425421473256086740184849, −8.053332941836933794835071931141, −6.84688861420732388851889155086, −6.43799960027812843772133501189, −5.74815106879120863847726169245, −4.34689992847520619663938618989, −2.78916696004872636435895102407, −2.20126072559128790780914950555,
0.25075513021654683670071817197, 1.61279326982373674786178753471, 4.12271086511942539694695461102, 4.37020277821051974463667656736, 4.90787387644438110303675372296, 6.49705485762284408136101888516, 7.04945482230910765600215184765, 8.480424104179659648366391737067, 9.131896140778575015303422557001, 9.749521410509147310891994103515
