L(s) = 1 | + (0.444 + 1.34i)2-s + (−1.60 + 1.19i)4-s + (1.00 − 0.579i)5-s + (−0.250 − 2.63i)7-s + (−2.31 − 1.62i)8-s + (1.22 + 1.08i)10-s + (1.17 + 0.679i)11-s + 3.61i·13-s + (3.42 − 1.50i)14-s + (1.14 − 3.83i)16-s + (6.02 + 3.48i)17-s + (3.04 + 5.27i)19-s + (−0.917 + 2.12i)20-s + (−0.388 + 1.88i)22-s + (2.38 − 1.37i)23-s + ⋯ |
L(s) = 1 | + (0.314 + 0.949i)2-s + (−0.802 + 0.597i)4-s + (0.448 − 0.258i)5-s + (−0.0948 − 0.995i)7-s + (−0.819 − 0.573i)8-s + (0.386 + 0.344i)10-s + (0.355 + 0.204i)11-s + 1.00i·13-s + (0.915 − 0.403i)14-s + (0.286 − 0.957i)16-s + (1.46 + 0.844i)17-s + (0.698 + 1.20i)19-s + (−0.205 + 0.475i)20-s + (−0.0828 + 0.401i)22-s + (0.498 − 0.287i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.184 - 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.40241 + 1.16365i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.40241 + 1.16365i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.444 - 1.34i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (0.250 + 2.63i)T \) |
good | 5 | \( 1 + (-1.00 + 0.579i)T + (2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-1.17 - 0.679i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 3.61iT - 13T^{2} \) |
| 17 | \( 1 + (-6.02 - 3.48i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.04 - 5.27i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.38 + 1.37i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 5.87T + 29T^{2} \) |
| 31 | \( 1 + (-2.09 + 3.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.83 - 3.17i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 5.72iT - 41T^{2} \) |
| 43 | \( 1 - 5.96iT - 43T^{2} \) |
| 47 | \( 1 + (6.26 + 10.8i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-4.95 + 8.57i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-0.862 + 1.49i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2.23 - 1.28i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.37 + 3.68i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 15.8iT - 71T^{2} \) |
| 73 | \( 1 + (3.53 + 2.04i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (0.0869 - 0.0502i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 3.12T + 83T^{2} \) |
| 89 | \( 1 + (-2.78 + 1.61i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 1.23iT - 97T^{2} \) |
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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
−10.02933236176758226224038898762, −9.827079382170149978866378483550, −8.621366392538333134086724432051, −7.82648808191720719152742815977, −6.99453805299817240903491814370, −6.17895334910250782675171412534, −5.27412610879449936096507566260, −4.20380085808199901931931157167, −3.43204186347910154502678816560, −1.33715953648965224949913939281,
1.05648691155342622498166105173, 2.73073843907995619600607565490, 3.12204244102426866457381092730, 4.78070494486077456407517283551, 5.51259863602002685908438496394, 6.30628443751869821224494609936, 7.70699140380570247845470452227, 8.776584013884558338845162194126, 9.503347896701505800585603273138, 10.14466340356904923053913410476