L(s) = 1 | + (0.949 − 1.64i)2-s + (2.65 + 1.40i)3-s + (0.196 + 0.340i)4-s + (0.399 + 4.98i)5-s + (4.82 − 3.02i)6-s + (−4.60 − 5.26i)7-s + 8.34·8-s + (5.04 + 7.45i)9-s + (8.57 + 4.07i)10-s + (−8.35 + 4.82i)11-s + (0.0423 + 1.17i)12-s − 16.9i·13-s + (−13.0 + 2.57i)14-s + (−5.94 + 13.7i)15-s + (7.13 − 12.3i)16-s + (−12.1 − 21.1i)17-s + ⋯ |
L(s) = 1 | + (0.474 − 0.822i)2-s + (0.883 + 0.468i)3-s + (0.0490 + 0.0850i)4-s + (0.0799 + 0.996i)5-s + (0.804 − 0.504i)6-s + (−0.658 − 0.752i)7-s + 1.04·8-s + (0.560 + 0.827i)9-s + (0.857 + 0.407i)10-s + (−0.759 + 0.438i)11-s + (0.00353 + 0.0981i)12-s − 1.30i·13-s + (−0.931 + 0.184i)14-s + (−0.396 + 0.918i)15-s + (0.446 − 0.772i)16-s + (−0.716 − 1.24i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.988 + 0.148i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(2.19468 - 0.163754i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.19468 - 0.163754i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-2.65 - 1.40i)T \) |
| 5 | \( 1 + (-0.399 - 4.98i)T \) |
| 7 | \( 1 + (4.60 + 5.26i)T \) |
good | 2 | \( 1 + (-0.949 + 1.64i)T + (-2 - 3.46i)T^{2} \) |
| 11 | \( 1 + (8.35 - 4.82i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 16.9iT - 169T^{2} \) |
| 17 | \( 1 + (12.1 + 21.1i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-6.95 + 12.0i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (0.354 - 0.614i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 - 16.5iT - 841T^{2} \) |
| 31 | \( 1 + (-7.12 - 12.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (1.08 + 0.626i)T + (684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + 24.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 57.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (16.0 - 27.8i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-8.67 - 15.0i)T + (-1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (75.8 - 43.8i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-52.2 + 90.4i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-86.1 + 49.7i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 + 50.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (81.5 - 47.0i)T + (2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (3.71 - 6.43i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + 69.6T + 6.88e3T^{2} \) |
| 89 | \( 1 + (-78.3 - 45.2i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + 90.4iT - 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.44816549454241828854165849297, −12.74195418452380776173270769230, −11.11747791796493526856962279146, −10.46066037249972862241507410491, −9.631079360291896521288544024019, −7.79941973066969503609195794003, −7.06026938854340389526027809473, −4.78278408134890098397866452442, −3.30834957273840650663355511119, −2.66125117462803068547134555250,
1.95696981453703377473036784161, 4.11946729931104834648632794084, 5.67648714672709117777227426704, 6.64362508531797206542521307318, 8.045864308849611478340429677565, 8.893574084359446267075408203816, 10.03787770802063919260472344353, 11.83775331206096983829080858055, 13.00350670535452213555197251783, 13.50421473769890785499942212903