L(s) = 1 | + (1.69 + 0.359i)3-s + (−1.74 − 3.01i)5-s + (−1.34 + 2.33i)7-s + (2.74 + 1.21i)9-s + (−2.84 + 4.92i)11-s + (1.76 + 3.04i)13-s + (−1.86 − 5.74i)15-s + 7.65·17-s − 2.02·19-s + (−3.12 + 3.47i)21-s + (−0.0370 − 0.0642i)23-s + (−3.57 + 6.18i)25-s + (4.20 + 3.05i)27-s + (2.46 − 4.26i)29-s + (3.72 + 6.44i)31-s + ⋯ |
L(s) = 1 | + (0.978 + 0.207i)3-s + (−0.779 − 1.34i)5-s + (−0.510 + 0.883i)7-s + (0.913 + 0.406i)9-s + (−0.857 + 1.48i)11-s + (0.488 + 0.845i)13-s + (−0.481 − 1.48i)15-s + 1.85·17-s − 0.463·19-s + (−0.682 + 0.758i)21-s + (−0.00773 − 0.0133i)23-s + (−0.714 + 1.23i)25-s + (0.809 + 0.587i)27-s + (0.456 − 0.791i)29-s + (0.668 + 1.15i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.558 - 0.829i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.785168044\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.785168044\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.69 - 0.359i)T \) |
good | 5 | \( 1 + (1.74 + 3.01i)T + (-2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (1.34 - 2.33i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (2.84 - 4.92i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.76 - 3.04i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 7.65T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 + (0.0370 + 0.0642i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.46 + 4.26i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-3.72 - 6.44i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 5.00T + 37T^{2} \) |
| 41 | \( 1 + (-0.482 - 0.834i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.255 - 0.442i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (2.83 - 4.91i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + (4.47 + 7.75i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.46 - 2.52i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.56 - 2.71i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 8.19T + 71T^{2} \) |
| 73 | \( 1 - 5.21T + 73T^{2} \) |
| 79 | \( 1 + (0.716 - 1.24i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.74 + 3.01i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 12.4T + 89T^{2} \) |
| 97 | \( 1 + (3.50 - 6.06i)T + (-48.5 - 84.0i)T^{2} \) |
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
−9.690978028867141963892152235601, −9.079862081665431371986951826129, −8.272120207368193421607667598811, −7.82779081269859016920325202223, −6.78854800246452982036337803182, −5.37857349414133791407750957898, −4.66622862986612151675937460129, −3.83260103493919785273793277144, −2.68159568090414750021172175433, −1.47649193283830367445833744345,
0.74252212524369723044314252658, 2.78242563553512050349387634253, 3.35171164477291646405854126898, 3.84232987020745014537597405318, 5.57628098589431956952615443908, 6.57074896431781787655515384387, 7.37365503073350901951252914166, 7.978309128843407594388645299245, 8.512263092526358025937439758926, 9.944447817922680310042048430854