L(s) = 1 | + (−0.834 − 1.14i)2-s + (−1.27 − 1.17i)3-s + (−0.608 + 1.90i)4-s + 1.08i·5-s + (−0.274 + 2.43i)6-s + (2.19 + 1.48i)7-s + (2.68 − 0.894i)8-s + (0.252 + 2.98i)9-s + (1.24 − 0.906i)10-s − 1.50·11-s + (3.00 − 1.71i)12-s + (1.59 + 2.75i)13-s + (−0.131 − 3.73i)14-s + (1.27 − 1.38i)15-s + (−3.26 − 2.31i)16-s + (5.40 − 3.11i)17-s + ⋯ |
L(s) = 1 | + (−0.589 − 0.807i)2-s + (−0.736 − 0.676i)3-s + (−0.304 + 0.952i)4-s + 0.486i·5-s + (−0.112 + 0.993i)6-s + (0.827 + 0.561i)7-s + (0.948 − 0.316i)8-s + (0.0842 + 0.996i)9-s + (0.392 − 0.286i)10-s − 0.453·11-s + (0.868 − 0.495i)12-s + (0.441 + 0.764i)13-s + (−0.0352 − 0.999i)14-s + (0.328 − 0.357i)15-s + (−0.815 − 0.579i)16-s + (1.31 − 0.756i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.917 + 0.396i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.917 + 0.396i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.763977 - 0.157994i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.763977 - 0.157994i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.834 + 1.14i)T \) |
| 3 | \( 1 + (1.27 + 1.17i)T \) |
| 7 | \( 1 + (-2.19 - 1.48i)T \) |
good | 5 | \( 1 - 1.08iT - 5T^{2} \) |
| 11 | \( 1 + 1.50T + 11T^{2} \) |
| 13 | \( 1 + (-1.59 - 2.75i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-5.40 + 3.11i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.29 - 1.32i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 3.25T + 23T^{2} \) |
| 29 | \( 1 + (1.78 + 1.03i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-0.708 - 0.409i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.35 - 7.55i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.57 + 3.79i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-3.95 - 2.28i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.81 - 11.7i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.686 - 0.396i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.19 - 2.07i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.36 + 4.09i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (7.96 + 4.59i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 + (4.02 + 6.97i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-12.4 + 7.21i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.40 - 4.16i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (5.90 + 3.41i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.69 + 11.5i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.91182094396174307424930522476, −11.12010917815560637819381963471, −10.37221776395246408617370579252, −9.177759920787238684037898804100, −7.960229786283515373755207826178, −7.34632539848247209666185395210, −5.90212902389155998287055048283, −4.66024435485787807997846645770, −2.83773662751352612475954678180, −1.44394212634052972460503980318,
0.996054677476280045566641163843, 4.00372036742903193273155742171, 5.22711048418530754863653520042, 5.77768446959949301850223439307, 7.27858780333681293958108717699, 8.172418132469607037302571201402, 9.164147229018219689677445006886, 10.38744076040722614428593022264, 10.66022042040783672441125793718, 11.92250241708472206532557515385