L(s) = 1 | + i·5-s + 1.73·7-s − 4.37i·11-s + 4.08i·13-s + 2.47i·17-s + (−4.12 − 1.42i)19-s − 3.66i·23-s − 25-s + 1.18·29-s + 0.741i·31-s + 1.73i·35-s − 8.33i·37-s + 6.48·41-s − 12.2·43-s − 6.85i·47-s + ⋯ |
L(s) = 1 | + 0.447i·5-s + 0.653·7-s − 1.31i·11-s + 1.13i·13-s + 0.600i·17-s + (−0.945 − 0.325i)19-s − 0.763i·23-s − 0.200·25-s + 0.219·29-s + 0.133i·31-s + 0.292i·35-s − 1.37i·37-s + 1.01·41-s − 1.87·43-s − 0.999i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.583 + 0.811i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.760897635\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.760897635\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 19 | \( 1 + (4.12 + 1.42i)T \) |
good | 7 | \( 1 - 1.73T + 7T^{2} \) |
| 11 | \( 1 + 4.37iT - 11T^{2} \) |
| 13 | \( 1 - 4.08iT - 13T^{2} \) |
| 17 | \( 1 - 2.47iT - 17T^{2} \) |
| 23 | \( 1 + 3.66iT - 23T^{2} \) |
| 29 | \( 1 - 1.18T + 29T^{2} \) |
| 31 | \( 1 - 0.741iT - 31T^{2} \) |
| 37 | \( 1 + 8.33iT - 37T^{2} \) |
| 41 | \( 1 - 6.48T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 6.85iT - 47T^{2} \) |
| 53 | \( 1 - 14.1T + 53T^{2} \) |
| 59 | \( 1 - 10.4T + 59T^{2} \) |
| 61 | \( 1 - 6.46T + 61T^{2} \) |
| 67 | \( 1 + 7.60iT - 67T^{2} \) |
| 71 | \( 1 + 14.3T + 71T^{2} \) |
| 73 | \( 1 - 10.4T + 73T^{2} \) |
| 79 | \( 1 + 12.1iT - 79T^{2} \) |
| 83 | \( 1 - 1.43iT - 83T^{2} \) |
| 89 | \( 1 + 0.714T + 89T^{2} \) |
| 97 | \( 1 + 2.31iT - 97T^{2} \) |
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
−7.962275776032756860951289871345, −7.04211831116452288797916365999, −6.49713017129178824139861052150, −5.84868056028047262111570987533, −5.01281549918572778642900598620, −4.15451971563374694179429727295, −3.58192753569016154146715826206, −2.48259277534832313512808313640, −1.79683268767689483714862425286, −0.48306206918415975065508991326,
0.986993976132878613721730368171, 1.91414120330302519509839550189, 2.76080182898155409313247661585, 3.83271686425599718986432151191, 4.61445119076848546936204442202, 5.13627033483649455316074974582, 5.83034374552117563833686411506, 6.81459505316331249563702340770, 7.40280434125160499461953482974, 8.175289316642091937953970079970