L(s) = 1 | + (0.781 + 0.623i)2-s + (1.29 − 2.68i)3-s + (0.222 + 0.974i)4-s + (−2.17 − 0.537i)5-s + (2.68 − 1.29i)6-s + (−2.64 + 0.0897i)7-s + (−0.433 + 0.900i)8-s + (−3.66 − 4.60i)9-s + (−1.36 − 1.77i)10-s + (2.95 − 3.70i)11-s + (2.90 + 0.663i)12-s + (−4.56 − 3.64i)13-s + (−2.12 − 1.57i)14-s + (−4.25 + 5.13i)15-s + (−0.900 + 0.433i)16-s + (−0.258 − 0.0588i)17-s + ⋯ |
L(s) = 1 | + (0.552 + 0.440i)2-s + (0.746 − 1.55i)3-s + (0.111 + 0.487i)4-s + (−0.970 − 0.240i)5-s + (1.09 − 0.527i)6-s + (−0.999 + 0.0339i)7-s + (−0.153 + 0.318i)8-s + (−1.22 − 1.53i)9-s + (−0.430 − 0.560i)10-s + (0.890 − 1.11i)11-s + (0.838 + 0.191i)12-s + (−1.26 − 1.01i)13-s + (−0.567 − 0.421i)14-s + (−1.09 + 1.32i)15-s + (−0.225 + 0.108i)16-s + (−0.0625 − 0.0142i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 490 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.298 + 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.980203 - 1.33379i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.980203 - 1.33379i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 5 | \( 1 + (2.17 + 0.537i)T \) |
| 7 | \( 1 + (2.64 - 0.0897i)T \) |
good | 3 | \( 1 + (-1.29 + 2.68i)T + (-1.87 - 2.34i)T^{2} \) |
| 11 | \( 1 + (-2.95 + 3.70i)T + (-2.44 - 10.7i)T^{2} \) |
| 13 | \( 1 + (4.56 + 3.64i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (0.258 + 0.0588i)T + (15.3 + 7.37i)T^{2} \) |
| 19 | \( 1 - 3.26T + 19T^{2} \) |
| 23 | \( 1 + (-8.34 + 1.90i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (1.42 - 6.24i)T + (-26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 - 2.52T + 31T^{2} \) |
| 37 | \( 1 + (1.26 + 0.288i)T + (33.3 + 16.0i)T^{2} \) |
| 41 | \( 1 + (-4.33 - 2.08i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-1.66 - 3.45i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (7.48 + 5.97i)T + (10.4 + 45.8i)T^{2} \) |
| 53 | \( 1 + (8.11 - 1.85i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-0.372 + 0.179i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-1.45 + 6.37i)T + (-54.9 - 26.4i)T^{2} \) |
| 67 | \( 1 - 1.23iT - 67T^{2} \) |
| 71 | \( 1 + (-0.298 - 1.30i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (-8.27 + 6.59i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 - 9.89T + 79T^{2} \) |
| 83 | \( 1 + (6.43 - 5.13i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-3.72 - 4.66i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 + 10.1iT - 97T^{2} \) |
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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
−11.08830041788437175345265240799, −9.367054591156153107205683381366, −8.597916390345517665997955026053, −7.75947102511775302595258228152, −7.05603299259698300953134224192, −6.40676390801178648620271951585, −5.11815432658481851856812910738, −3.36258469624463597027958182536, −2.96476340684490942382524697100, −0.77843131803667922993394344401,
2.59700116761247498275765596050, 3.52253632025376921395040531919, 4.31014729903311570538056578463, 4.95372572670224151781864624342, 6.68869260708886300658066463122, 7.53932990957603013904261473982, 9.126697581305027221260834208792, 9.487257029290654419573447249369, 10.17491439056925113408297032263, 11.25539437953922063607544105145