L(s) = 1 | + (−2.36 + 1.36i)3-s + (−0.921 + 1.59i)5-s + (−0.396 + 2.61i)7-s + (2.23 − 3.86i)9-s + (2.51 − 2.16i)11-s − 6.17i·13-s − 5.03i·15-s + (4.14 − 2.39i)17-s + (0.0381 − 0.0661i)19-s + (−2.63 − 6.73i)21-s + (2.14 + 1.24i)23-s + (0.800 + 1.38i)25-s + 4.01i·27-s − 7.13i·29-s + (−4.75 + 2.74i)31-s + ⋯ |
L(s) = 1 | + (−1.36 + 0.788i)3-s + (−0.412 + 0.714i)5-s + (−0.149 + 0.988i)7-s + (0.744 − 1.28i)9-s + (0.757 − 0.652i)11-s − 1.71i·13-s − 1.30i·15-s + (1.00 − 0.580i)17-s + (0.00875 − 0.0151i)19-s + (−0.575 − 1.46i)21-s + (0.448 + 0.258i)23-s + (0.160 + 0.277i)25-s + 0.771i·27-s − 1.32i·29-s + (−0.853 + 0.492i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.811 - 0.584i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9035081790\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9035081790\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (0.396 - 2.61i)T \) |
| 11 | \( 1 + (-2.51 + 2.16i)T \) |
good | 3 | \( 1 + (2.36 - 1.36i)T + (1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.921 - 1.59i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 6.17iT - 13T^{2} \) |
| 17 | \( 1 + (-4.14 + 2.39i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.0381 + 0.0661i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.14 - 1.24i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.13iT - 29T^{2} \) |
| 31 | \( 1 + (4.75 - 2.74i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.83 + 6.63i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 9.30iT - 41T^{2} \) |
| 43 | \( 1 - 6.89T + 43T^{2} \) |
| 47 | \( 1 + (-0.725 - 0.419i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.492 - 0.852i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (2.20 - 1.27i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-6.50 - 3.75i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.06 - 3.49i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12.3iT - 71T^{2} \) |
| 73 | \( 1 + (-8.99 + 5.19i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.83 + 3.17i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 4.42T + 83T^{2} \) |
| 89 | \( 1 + (0.870 - 1.50i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 4.56T + 97T^{2} \) |
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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
−9.865097686111367599590225380652, −9.290217701369032124043740411170, −8.114068131765848164731310791654, −7.23770342732502370662595549054, −5.99891697396125486716308541375, −5.78527050862355056509587255296, −4.88815130821970384877850586380, −3.63124326680035281239452339688, −2.89888450996068346491898139969, −0.69277277062926079834053643476,
0.884842894773285185626511571375, 1.70309952231484024348313206703, 3.85654024845099181066805580475, 4.52310604898458788872773525341, 5.43080629540659218177102646973, 6.56251965991597287036291808868, 6.93540903568708563685660525139, 7.70773440815544767740598687225, 8.843964049331607713648402631414, 9.683568782754049657775105633278