L(s) = 1 | + (−0.258 − 0.965i)2-s + (−0.965 + 0.258i)3-s + (−0.866 + 0.499i)4-s + (2.23 − 0.109i)5-s + (0.499 + 0.866i)6-s + (−0.613 − 0.613i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + (−0.683 − 2.12i)10-s − 3.43·11-s + (0.707 − 0.707i)12-s + (0.543 − 2.02i)13-s + (−0.434 + 0.751i)14-s + (−2.12 + 0.683i)15-s + (0.500 − 0.866i)16-s + (5.79 − 1.55i)17-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−0.557 + 0.149i)3-s + (−0.433 + 0.249i)4-s + (0.998 − 0.0489i)5-s + (0.204 + 0.353i)6-s + (−0.232 − 0.232i)7-s + (0.249 + 0.249i)8-s + (0.288 − 0.166i)9-s + (−0.216 − 0.673i)10-s − 1.03·11-s + (0.204 − 0.204i)12-s + (0.150 − 0.562i)13-s + (−0.116 + 0.200i)14-s + (−0.549 + 0.176i)15-s + (0.125 − 0.216i)16-s + (1.40 − 0.376i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 570 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.129 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.882214 - 0.774436i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.882214 - 0.774436i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 3 | \( 1 + (0.965 - 0.258i)T \) |
| 5 | \( 1 + (-2.23 + 0.109i)T \) |
| 19 | \( 1 + (-3.87 + 2.00i)T \) |
good | 7 | \( 1 + (0.613 + 0.613i)T + 7iT^{2} \) |
| 11 | \( 1 + 3.43T + 11T^{2} \) |
| 13 | \( 1 + (-0.543 + 2.02i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (-5.79 + 1.55i)T + (14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (-2.31 - 0.621i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (4.59 + 7.95i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 8.09iT - 31T^{2} \) |
| 37 | \( 1 + (-7.31 + 7.31i)T - 37iT^{2} \) |
| 41 | \( 1 + (6.91 + 3.99i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (0.968 + 3.61i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-1.89 + 7.08i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.887 - 3.31i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-3.68 + 6.37i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3.34 - 5.79i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.18 - 1.38i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (5.85 + 3.38i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.07 - 11.4i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (6.09 - 10.5i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.509 + 0.509i)T - 83iT^{2} \) |
| 89 | \( 1 + (-6.81 - 11.8i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.874 - 3.26i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−10.32770014429717787770265112486, −9.998148990026682602995933768783, −9.122395952456723965672393243502, −7.919112987946015587087853538342, −6.95965880289530189335927932124, −5.47385988667680265766914592639, −5.27981054607541660953601527182, −3.59314350272571368520346068441, −2.47671007147439845292972420282, −0.864231256633635259112675728647,
1.42278567051605193241184149337, 3.08920053303855974663535535631, 4.82769545314140582466135684677, 5.64285813605056473814810826348, 6.21603723178728061383154799807, 7.31766977250925483918306474089, 8.116646539378126863092529136524, 9.362214858649881120742300986089, 9.905561124477596596218066085649, 10.73495142206472512277879990229