L(s) = 1 | + (−0.342 − 0.939i)2-s + (0.326 + 0.118i)3-s + (−0.766 + 0.642i)4-s + (2.57 − 0.453i)5-s − 0.347i·6-s + (−0.361 − 2.04i)7-s + (0.866 + 0.500i)8-s + (−2.20 − 1.85i)9-s + (−1.30 − 2.26i)10-s + (−2.99 + 5.19i)11-s + (−0.326 + 0.118i)12-s + (2.64 + 3.15i)13-s + (−1.80 + 1.03i)14-s + (0.893 + 0.157i)15-s + (0.173 − 0.984i)16-s + (−0.618 + 0.737i)17-s + ⋯ |
L(s) = 1 | + (−0.241 − 0.664i)2-s + (0.188 + 0.0685i)3-s + (−0.383 + 0.321i)4-s + (1.15 − 0.202i)5-s − 0.141i·6-s + (−0.136 − 0.773i)7-s + (0.306 + 0.176i)8-s + (−0.735 − 0.616i)9-s + (−0.412 − 0.715i)10-s + (−0.903 + 1.56i)11-s + (−0.0942 + 0.0342i)12-s + (0.733 + 0.874i)13-s + (−0.481 + 0.277i)14-s + (0.230 + 0.0406i)15-s + (0.0434 − 0.246i)16-s + (−0.150 + 0.178i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.683 + 0.729i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.842591 - 0.365173i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.842591 - 0.365173i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.342 + 0.939i)T \) |
| 37 | \( 1 + (-6.07 + 0.383i)T \) |
good | 3 | \( 1 + (-0.326 - 0.118i)T + (2.29 + 1.92i)T^{2} \) |
| 5 | \( 1 + (-2.57 + 0.453i)T + (4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (0.361 + 2.04i)T + (-6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (2.99 - 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.64 - 3.15i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (0.618 - 0.737i)T + (-2.95 - 16.7i)T^{2} \) |
| 19 | \( 1 + (-0.534 + 1.46i)T + (-14.5 - 12.2i)T^{2} \) |
| 23 | \( 1 + (5.51 - 3.18i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.51 + 2.02i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.39iT - 31T^{2} \) |
| 41 | \( 1 + (-7.94 + 6.66i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + 3.76iT - 43T^{2} \) |
| 47 | \( 1 + (-3.08 - 5.34i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (1.39 - 7.90i)T + (-49.8 - 18.1i)T^{2} \) |
| 59 | \( 1 + (-5.02 - 0.885i)T + (55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (6.25 + 7.45i)T + (-10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (1.83 + 10.3i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (10.1 + 3.69i)T + (54.3 + 45.6i)T^{2} \) |
| 73 | \( 1 - 3.55T + 73T^{2} \) |
| 79 | \( 1 + (-2.51 + 0.442i)T + (74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-5.29 - 4.44i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (16.0 + 2.82i)T + (83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (14.1 - 8.15i)T + (48.5 - 84.0i)T^{2} \) |
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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
−14.05805349543429637179215936251, −13.40908159771612845373380319850, −12.31249833883572106235783952019, −10.97298420049924846231907112288, −9.817177415165528037415429575887, −9.216357267577790396664874122347, −7.57033113344179613471238587110, −5.92952947481102911955446889296, −4.16219124229754266687910926540, −2.13607921672580497508811577376,
2.74776678098280708047157717487, 5.65282805162515088236084507233, 5.93420191587257392533837633179, 8.035984789040003294085181929543, 8.771686294529636244560838956742, 10.15030202185912976102314786310, 11.15764598120730800311657781189, 13.03859763678394129810588830790, 13.73981841510577960413377916963, 14.61295628713703624918341568671