L(s) = 1 | + (−0.0204 + 1.41i)2-s + (0.290 + 1.92i)3-s + (−1.99 − 0.0577i)4-s + (1.10 + 0.434i)5-s + (−2.72 + 0.370i)6-s + (1.77 + 1.96i)7-s + (0.122 − 2.82i)8-s + (−0.752 + 0.232i)9-s + (−0.637 + 1.55i)10-s + (−1.15 + 3.74i)11-s + (−0.468 − 3.86i)12-s + (−4.65 + 1.06i)13-s + (−2.81 + 2.47i)14-s + (−0.515 + 2.25i)15-s + (3.99 + 0.230i)16-s + (2.11 − 1.44i)17-s + ⋯ |
L(s) = 1 | + (−0.0144 + 0.999i)2-s + (0.167 + 1.11i)3-s + (−0.999 − 0.0288i)4-s + (0.495 + 0.194i)5-s + (−1.11 + 0.151i)6-s + (0.671 + 0.741i)7-s + (0.0432 − 0.999i)8-s + (−0.250 + 0.0773i)9-s + (−0.201 + 0.492i)10-s + (−0.348 + 1.12i)11-s + (−0.135 − 1.11i)12-s + (−1.29 + 0.294i)13-s + (−0.751 + 0.660i)14-s + (−0.133 + 0.583i)15-s + (0.998 + 0.0576i)16-s + (0.512 − 0.349i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.994 - 0.100i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0693727 + 1.38328i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0693727 + 1.38328i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.0204 - 1.41i)T \) |
| 7 | \( 1 + (-1.77 - 1.96i)T \) |
good | 3 | \( 1 + (-0.290 - 1.92i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-1.10 - 0.434i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (1.15 - 3.74i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (4.65 - 1.06i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-2.11 + 1.44i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (-3.04 + 1.75i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (3.64 + 2.48i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-2.00 + 4.16i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (-1.49 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.67 - 0.125i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (-1.46 - 1.83i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-6.23 - 4.97i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-8.15 + 7.56i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (-0.517 - 0.0387i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (9.32 - 3.66i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (3.74 - 0.280i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (-9.58 - 5.53i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (7.89 - 3.80i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-1.56 - 1.45i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-6.40 - 11.0i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.80 - 1.32i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-7.80 + 2.40i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−11.86371357132259892877950973254, −10.29564761307470317777585422471, −9.761972167889024795072429305785, −9.202070309575813521449799100497, −7.976694883774756651946357740822, −7.16265782284890276227929908166, −5.82302550698623449501981552871, −4.90095168828932534120149606338, −4.30934046605671642607790540734, −2.47745022258447110716540379752,
0.984406043999087072706975755993, 2.08125358672085832643016129684, 3.43533677602527199562408431250, 4.91318342406124717444869199381, 5.88615658108922176214157645170, 7.56324096699346203870751585641, 7.895672284504080889580748641182, 9.126952441389624777934654750949, 10.17705019203844200373449123484, 10.82954480628089439117633010532