| L(s) = 1 | + (0.0747 − 0.997i)2-s + (−1.51 + 0.831i)3-s + (−0.988 − 0.149i)4-s + (−3.10 − 2.11i)5-s + (0.715 + 1.57i)6-s + (1.34 − 0.202i)7-s + (−0.222 + 0.974i)8-s + (1.61 − 2.52i)9-s + (−2.34 + 2.93i)10-s + (−0.998 − 0.308i)11-s + (1.62 − 0.595i)12-s + (−1.51 + 1.41i)13-s + (−0.101 − 1.35i)14-s + (6.47 + 0.635i)15-s + (0.955 + 0.294i)16-s + 2.33·17-s + ⋯ |
| L(s) = 1 | + (0.0528 − 0.705i)2-s + (−0.877 + 0.480i)3-s + (−0.494 − 0.0745i)4-s + (−1.38 − 0.946i)5-s + (0.292 + 0.643i)6-s + (0.508 − 0.0765i)7-s + (−0.0786 + 0.344i)8-s + (0.539 − 0.842i)9-s + (−0.740 + 0.929i)10-s + (−0.301 − 0.0929i)11-s + (0.469 − 0.171i)12-s + (−0.421 + 0.391i)13-s + (−0.0271 − 0.362i)14-s + (1.67 + 0.164i)15-s + (0.238 + 0.0736i)16-s + 0.565·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.413 - 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.316854 + 0.204207i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.316854 + 0.204207i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.0747 + 0.997i)T \) |
| 3 | \( 1 + (1.51 - 0.831i)T \) |
| 29 | \( 1 + (-0.988 + 5.29i)T \) |
| good | 5 | \( 1 + (3.10 + 2.11i)T + (1.82 + 4.65i)T^{2} \) |
| 7 | \( 1 + (-1.34 + 0.202i)T + (6.68 - 2.06i)T^{2} \) |
| 11 | \( 1 + (0.998 + 0.308i)T + (9.08 + 6.19i)T^{2} \) |
| 13 | \( 1 + (1.51 - 1.41i)T + (0.971 - 12.9i)T^{2} \) |
| 17 | \( 1 - 2.33T + 17T^{2} \) |
| 19 | \( 1 + (5.34 - 6.70i)T + (-4.22 - 18.5i)T^{2} \) |
| 23 | \( 1 + (-0.347 - 4.63i)T + (-22.7 + 3.42i)T^{2} \) |
| 31 | \( 1 + (-8.31 - 5.66i)T + (11.3 + 28.8i)T^{2} \) |
| 37 | \( 1 + (0.0929 - 0.407i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (-3.48 - 6.03i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.38 + 0.944i)T + (15.7 - 40.0i)T^{2} \) |
| 47 | \( 1 + (9.65 + 2.97i)T + (38.8 + 26.4i)T^{2} \) |
| 53 | \( 1 + (8.96 - 4.31i)T + (33.0 - 41.4i)T^{2} \) |
| 59 | \( 1 + (-4.02 - 6.96i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4.42 - 0.666i)T + (58.2 - 17.9i)T^{2} \) |
| 67 | \( 1 + (-8.03 + 2.47i)T + (55.3 - 37.7i)T^{2} \) |
| 71 | \( 1 + (3.54 + 15.5i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (4.72 + 2.27i)T + (45.5 + 57.0i)T^{2} \) |
| 79 | \( 1 + (-3.96 - 3.68i)T + (5.90 + 78.7i)T^{2} \) |
| 83 | \( 1 + (-0.0269 - 0.0686i)T + (-60.8 + 56.4i)T^{2} \) |
| 89 | \( 1 + (4.74 - 2.28i)T + (55.4 - 69.5i)T^{2} \) |
| 97 | \( 1 + (-2.36 - 6.03i)T + (-71.1 + 65.9i)T^{2} \) |
| 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
−11.19727819547357660184742218106, −10.33175949706392349978661278459, −9.475889827339287152224694632657, −8.308578347952382194773666916723, −7.76313864594437895897238358306, −6.20482756461826900422756970238, −4.95413072461001734292406429824, −4.42718652611879609355881152345, −3.50862539946979853826033775847, −1.30333067445058879479702849941,
0.27365664836520202421107152651, 2.73049060086280770120268672322, 4.31825011518582400230146853740, 5.02872633969712809268307740778, 6.41684557593802526969151196168, 6.98353357892538228604021228212, 7.84440699550005694882672445613, 8.401614143446989784829972286964, 10.05729613294095178736357051048, 10.96752605233006075081875231922
