L(s) = 1 | + (−0.345 + 1.37i)2-s + (−1.76 − 0.946i)4-s + (0.561 − 2.16i)5-s − 4.51·7-s + (1.90 − 2.08i)8-s + (2.77 + 1.51i)10-s + (3.44 + 3.44i)11-s + (0.113 + 0.113i)13-s + (1.55 − 6.19i)14-s + (2.20 + 3.33i)16-s + 5.03i·17-s + (−0.992 + 0.992i)19-s + (−3.03 + 3.28i)20-s + (−5.90 + 3.53i)22-s − 8.00·23-s + ⋯ |
L(s) = 1 | + (−0.244 + 0.969i)2-s + (−0.880 − 0.473i)4-s + (0.251 − 0.967i)5-s − 1.70·7-s + (0.673 − 0.738i)8-s + (0.877 + 0.479i)10-s + (1.03 + 1.03i)11-s + (0.0315 + 0.0315i)13-s + (0.416 − 1.65i)14-s + (0.551 + 0.833i)16-s + 1.22i·17-s + (−0.227 + 0.227i)19-s + (−0.679 + 0.733i)20-s + (−1.25 + 0.752i)22-s − 1.66·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 - 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0132169 + 0.426765i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0132169 + 0.426765i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.345 - 1.37i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-0.561 + 2.16i)T \) |
good | 7 | \( 1 + 4.51T + 7T^{2} \) |
| 11 | \( 1 + (-3.44 - 3.44i)T + 11iT^{2} \) |
| 13 | \( 1 + (-0.113 - 0.113i)T + 13iT^{2} \) |
| 17 | \( 1 - 5.03iT - 17T^{2} \) |
| 19 | \( 1 + (0.992 - 0.992i)T - 19iT^{2} \) |
| 23 | \( 1 + 8.00T + 23T^{2} \) |
| 29 | \( 1 + (-1.01 + 1.01i)T - 29iT^{2} \) |
| 31 | \( 1 + 6.42T + 31T^{2} \) |
| 37 | \( 1 + (1.63 - 1.63i)T - 37iT^{2} \) |
| 41 | \( 1 - 3.35iT - 41T^{2} \) |
| 43 | \( 1 + (5.68 - 5.68i)T - 43iT^{2} \) |
| 47 | \( 1 - 9.10iT - 47T^{2} \) |
| 53 | \( 1 + (-3.27 + 3.27i)T - 53iT^{2} \) |
| 59 | \( 1 + (-5.30 - 5.30i)T + 59iT^{2} \) |
| 61 | \( 1 + (5.87 - 5.87i)T - 61iT^{2} \) |
| 67 | \( 1 + (-1.87 - 1.87i)T + 67iT^{2} \) |
| 71 | \( 1 + 0.635iT - 71T^{2} \) |
| 73 | \( 1 + 6.14T + 73T^{2} \) |
| 79 | \( 1 - 1.76T + 79T^{2} \) |
| 83 | \( 1 + (6.39 + 6.39i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.579iT - 89T^{2} \) |
| 97 | \( 1 + 15.2iT - 97T^{2} \) |
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
−10.20272561181714120871864960602, −9.811032105007700241166163118872, −9.115879510820936889111876596905, −8.333059227788376592251849198802, −7.25463540527916886549770909299, −6.25895675412742427245783894618, −5.91168657745499733714441451047, −4.45213891875160410407251474851, −3.77069054803586022581321469481, −1.62013692475495836549878740465,
0.23863433213088286075219553251, 2.24636418576468631955158661812, 3.36347343360455087967066752662, 3.77337592845243245063266979624, 5.57720983054961636587770016110, 6.49879852694152929597096135363, 7.27930887873626706762600626144, 8.653616723312224564453688625138, 9.403430327271716334368101740632, 9.995213953526529410737664417013