L(s) = 1 | + (0.649 − 1.25i)2-s + (0.639 − 0.639i)3-s + (−1.15 − 1.63i)4-s + (1.45 + 1.69i)5-s + (−0.388 − 1.21i)6-s + (0.0280 − 2.64i)7-s + (−2.80 + 0.395i)8-s + 2.18i·9-s + (3.07 − 0.728i)10-s − 5.47i·11-s + (−1.78 − 0.303i)12-s + (3.89 − 3.89i)13-s + (−3.30 − 1.75i)14-s + (2.01 + 0.153i)15-s + (−1.32 + 3.77i)16-s + (−3.30 + 3.30i)17-s + ⋯ |
L(s) = 1 | + (0.459 − 0.888i)2-s + (0.369 − 0.369i)3-s + (−0.578 − 0.815i)4-s + (0.651 + 0.758i)5-s + (−0.158 − 0.497i)6-s + (0.0106 − 0.999i)7-s + (−0.990 + 0.139i)8-s + 0.727i·9-s + (0.973 − 0.230i)10-s − 1.65i·11-s + (−0.514 − 0.0875i)12-s + (1.07 − 1.07i)13-s + (−0.883 − 0.468i)14-s + (0.520 + 0.0396i)15-s + (−0.330 + 0.943i)16-s + (−0.801 + 0.801i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.177 + 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.16458 - 1.39307i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.16458 - 1.39307i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.649 + 1.25i)T \) |
| 5 | \( 1 + (-1.45 - 1.69i)T \) |
| 7 | \( 1 + (-0.0280 + 2.64i)T \) |
good | 3 | \( 1 + (-0.639 + 0.639i)T - 3iT^{2} \) |
| 11 | \( 1 + 5.47iT - 11T^{2} \) |
| 13 | \( 1 + (-3.89 + 3.89i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.30 - 3.30i)T - 17iT^{2} \) |
| 19 | \( 1 - 5.24iT - 19T^{2} \) |
| 23 | \( 1 + (0.137 - 0.137i)T - 23iT^{2} \) |
| 29 | \( 1 + 2.31T + 29T^{2} \) |
| 31 | \( 1 - 4.44iT - 31T^{2} \) |
| 37 | \( 1 + (-4.36 + 4.36i)T - 37iT^{2} \) |
| 41 | \( 1 - 4.27iT - 41T^{2} \) |
| 43 | \( 1 + (-1.77 - 1.77i)T + 43iT^{2} \) |
| 47 | \( 1 + (0.164 - 0.164i)T - 47iT^{2} \) |
| 53 | \( 1 + (-2.76 - 2.76i)T + 53iT^{2} \) |
| 59 | \( 1 + 2.55iT - 59T^{2} \) |
| 61 | \( 1 + 0.863T + 61T^{2} \) |
| 67 | \( 1 + (4.38 - 4.38i)T - 67iT^{2} \) |
| 71 | \( 1 + 3.84T + 71T^{2} \) |
| 73 | \( 1 + (-8.74 - 8.74i)T + 73iT^{2} \) |
| 79 | \( 1 + 14.5iT - 79T^{2} \) |
| 83 | \( 1 + (0.395 - 0.395i)T - 83iT^{2} \) |
| 89 | \( 1 + 16.2T + 89T^{2} \) |
| 97 | \( 1 + (-11.0 + 11.0i)T - 97iT^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.20651738522011306081347817903, −10.77149302121301855475010960156, −10.20880234895941803750620325077, −8.744097757557562005794031227010, −7.87949190333528147211809173680, −6.32699456163457323180624725968, −5.60516752643558245388187473020, −3.84481529374059995511557689597, −2.96604133755799636015749813785, −1.42592643414438673810186144819,
2.38741557311574729875579550350, 4.18147330470203280287188167003, 4.93360125769795746643151689005, 6.17186252945669838305015743227, 6.98708130723491300385820423204, 8.510781176001352294296849896687, 9.282707442865158002689129315042, 9.500565578215089528955780581244, 11.54687077460869283427444864656, 12.32559653912523112172837884329