L(s) = 1 | + (−0.173 − 0.984i)2-s + (−0.0299 + 0.0251i)3-s + (−0.939 + 0.342i)4-s + (−2.55 − 0.931i)5-s + (0.0299 + 0.0251i)6-s + (−0.483 + 0.838i)7-s + (0.5 + 0.866i)8-s + (−0.520 + 2.95i)9-s + (−0.472 + 2.68i)10-s + (0.5 + 0.866i)11-s + (0.0195 − 0.0338i)12-s + (4.98 + 4.18i)13-s + (0.909 + 0.331i)14-s + (0.100 − 0.0364i)15-s + (0.766 − 0.642i)16-s + (−0.597 − 3.38i)17-s + ⋯ |
L(s) = 1 | + (−0.122 − 0.696i)2-s + (−0.0173 + 0.0145i)3-s + (−0.469 + 0.171i)4-s + (−1.14 − 0.416i)5-s + (0.0122 + 0.0102i)6-s + (−0.182 + 0.316i)7-s + (0.176 + 0.306i)8-s + (−0.173 + 0.984i)9-s + (−0.149 + 0.847i)10-s + (0.150 + 0.261i)11-s + (0.00564 − 0.00978i)12-s + (1.38 + 1.16i)13-s + (0.243 + 0.0884i)14-s + (0.0258 − 0.00940i)15-s + (0.191 − 0.160i)16-s + (−0.144 − 0.821i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 418 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.761 - 0.648i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.758206 + 0.279235i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.758206 + 0.279235i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 + 0.984i)T \) |
| 11 | \( 1 + (-0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.721 - 4.29i)T \) |
good | 3 | \( 1 + (0.0299 - 0.0251i)T + (0.520 - 2.95i)T^{2} \) |
| 5 | \( 1 + (2.55 + 0.931i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (0.483 - 0.838i)T + (-3.5 - 6.06i)T^{2} \) |
| 13 | \( 1 + (-4.98 - 4.18i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.597 + 3.38i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.88 + 1.05i)T + (17.6 - 14.7i)T^{2} \) |
| 29 | \( 1 + (0.984 - 5.58i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (2.33 - 4.05i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + 4.12T + 37T^{2} \) |
| 41 | \( 1 + (4.35 - 3.65i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (0.335 + 0.122i)T + (32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.38 - 7.86i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (1.16 - 0.425i)T + (40.6 - 34.0i)T^{2} \) |
| 59 | \( 1 + (1.76 + 10.0i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-4.82 + 1.75i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.647 + 3.67i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (-6.33 - 2.30i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 + (4.76 - 3.99i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-2.40 + 2.02i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.765 + 1.32i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-5.88 - 4.93i)T + (15.4 + 87.6i)T^{2} \) |
| 97 | \( 1 + (-2.76 - 15.6i)T + (-91.1 + 33.1i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.29475736253987494681468425940, −10.72090141689916889931362539388, −9.395443406181511852145794650478, −8.641499338369094297182911906658, −7.896807456098003722219183123266, −6.74760775563251520969236299478, −5.22900960888862106374577261487, −4.27226077662076248398513986184, −3.26009271747402924653704801946, −1.59873318390243628586384606682,
0.58671926849517229598829825449, 3.39593777795335715980912875840, 3.94973460716028843128515994332, 5.57994161125344379878586429738, 6.51065486170629361946756288778, 7.35458898302374296872905916015, 8.313694806880595348687900753791, 8.964586357982430508100387521925, 10.24702906502289343303399022525, 11.11450625006604597621814430252