L(s) = 1 | + (1.5 − 0.866i)3-s + (−1 − 1.73i)4-s + (−0.5 − 2.59i)7-s + (1.5 − 2.59i)9-s + (−3 − 1.73i)12-s + 1.73i·13-s + (−1.99 + 3.46i)16-s + (−4.5 − 2.59i)19-s + (−3 − 3.46i)21-s − 5.19i·27-s + (−4 + 3.46i)28-s + (7.5 − 4.33i)31-s − 6·36-s + (0.5 − 0.866i)37-s + (1.49 + 2.59i)39-s + ⋯ |
L(s) = 1 | + (0.866 − 0.499i)3-s + (−0.5 − 0.866i)4-s + (−0.188 − 0.981i)7-s + (0.5 − 0.866i)9-s + (−0.866 − 0.499i)12-s + 0.480i·13-s + (−0.499 + 0.866i)16-s + (−1.03 − 0.596i)19-s + (−0.654 − 0.755i)21-s − 0.999i·27-s + (−0.755 + 0.654i)28-s + (1.34 − 0.777i)31-s − 36-s + (0.0821 − 0.142i)37-s + (0.240 + 0.416i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.444 + 0.895i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.807315 - 1.30127i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.807315 - 1.30127i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (0.5 + 2.59i)T \) |
good | 2 | \( 1 + (1 + 1.73i)T^{2} \) |
| 11 | \( 1 + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 1.73iT - 13T^{2} \) |
| 17 | \( 1 + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.5 + 2.59i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + (-7.5 + 4.33i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.5 + 0.866i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-6 - 3.46i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.5 - 9.52i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 71T^{2} \) |
| 73 | \( 1 + (-13.5 + 7.79i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.5 + 11.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 13.8iT - 97T^{2} \) |
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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
−10.37650257749575262375681195705, −9.641709154406238161222607458249, −8.866237202378373240600551070480, −7.952681727012636919654588356996, −6.87867987815446069435897347762, −6.20240937006674184708994844485, −4.64585070953819687191436353598, −3.86116311364689938946201580604, −2.28147068853533253333925174862, −0.848079794222887665276495082755,
2.36153412573112177706269710039, 3.28687116431095061510459503434, 4.30830464324035628712094546409, 5.32965068511905031149613030409, 6.70720853015046547331354729237, 8.063727837778368209563984116739, 8.369414729250721848152321888788, 9.264492018743605027303909701158, 9.999621213421722491316626640805, 11.07819973987764280653510974994