L(s) = 1 | + (0.0473 + 0.176i)2-s + (0.866 + 0.5i)3-s + (1.70 − 0.983i)4-s + (−2.80 − 2.80i)5-s + (−0.0473 + 0.176i)6-s + (1.70 − 2.02i)7-s + (0.513 + 0.513i)8-s + (0.499 + 0.866i)9-s + (0.362 − 0.627i)10-s + (−2.53 + 0.679i)11-s + 1.96·12-s + (−1.37 − 3.33i)13-s + (0.437 + 0.205i)14-s + (−1.02 − 3.82i)15-s + (1.90 − 3.29i)16-s + (1.43 + 2.48i)17-s + ⋯ |
L(s) = 1 | + (0.0334 + 0.124i)2-s + (0.499 + 0.288i)3-s + (0.851 − 0.491i)4-s + (−1.25 − 1.25i)5-s + (−0.0193 + 0.0721i)6-s + (0.645 − 0.764i)7-s + (0.181 + 0.181i)8-s + (0.166 + 0.288i)9-s + (0.114 − 0.198i)10-s + (−0.764 + 0.204i)11-s + 0.567·12-s + (−0.380 − 0.924i)13-s + (0.117 + 0.0550i)14-s + (−0.264 − 0.987i)15-s + (0.475 − 0.822i)16-s + (0.347 + 0.601i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.38346 - 0.645824i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.38346 - 0.645824i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (-1.70 + 2.02i)T \) |
| 13 | \( 1 + (1.37 + 3.33i)T \) |
good | 2 | \( 1 + (-0.0473 - 0.176i)T + (-1.73 + i)T^{2} \) |
| 5 | \( 1 + (2.80 + 2.80i)T + 5iT^{2} \) |
| 11 | \( 1 + (2.53 - 0.679i)T + (9.52 - 5.5i)T^{2} \) |
| 17 | \( 1 + (-1.43 - 2.48i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.759 - 2.83i)T + (-16.4 - 9.5i)T^{2} \) |
| 23 | \( 1 + (-7.27 - 4.19i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.66 + 2.87i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-6.75 - 6.75i)T + 31iT^{2} \) |
| 37 | \( 1 + (6.77 - 1.81i)T + (32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + (-2.79 + 0.747i)T + (35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (2.43 - 1.40i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.85 + 4.85i)T - 47iT^{2} \) |
| 53 | \( 1 + 5.43T + 53T^{2} \) |
| 59 | \( 1 + (-0.00666 - 0.00178i)T + (51.0 + 29.5i)T^{2} \) |
| 61 | \( 1 + (5.65 - 3.26i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-2.10 - 7.84i)T + (-58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (14.5 + 3.91i)T + (61.4 + 35.5i)T^{2} \) |
| 73 | \( 1 + (0.321 - 0.321i)T - 73iT^{2} \) |
| 79 | \( 1 - 0.280T + 79T^{2} \) |
| 83 | \( 1 + (-2.42 - 2.42i)T + 83iT^{2} \) |
| 89 | \( 1 + (0.0536 + 0.200i)T + (-77.0 + 44.5i)T^{2} \) |
| 97 | \( 1 + (0.197 - 0.736i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−11.79499822752320411321848868751, −10.74455728374045875601457772094, −10.10061260223848409059075536829, −8.582531403698049715222504549393, −7.87482834460996523026385515739, −7.25153770418778867188390608042, −5.36473546601977310779265519189, −4.62994639309854519967271883522, −3.25722980957878214179057514875, −1.24594006811624680498311415849,
2.48230146333342206766419637493, 3.08061636087528941527474414927, 4.58573867433473192601746479448, 6.48324774157765388196642568594, 7.28394325004098756779027601389, 7.912734572711292966849139976948, 8.898772516511760928251467816102, 10.53736371948398211233561620967, 11.27552559909997553223473441853, 11.84618292705183707581948229948