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
−10.82954480628089439117633010532, −10.17705019203844200373449123484, −9.126952441389624777934654750949, −7.895672284504080889580748641182, −7.56324096699346203870751585641, −5.88615658108922176214157645170, −4.91318342406124717444869199381, −3.43533677602527199562408431250, −2.08125358672085832643016129684, −0.984406043999087072706975755993,
2.47745022258447110716540379752, 4.30934046605671642607790540734, 4.90095168828932534120149606338, 5.82302550698623449501981552871, 7.16265782284890276227929908166, 7.976694883774756651946357740822, 9.202070309575813521449799100497, 9.761972167889024795072429305785, 10.29564761307470317777585422471, 11.86371357132259892877950973254