| L(s) = 1 | + (−0.677 + 0.932i)2-s + (1.03 + 3.18i)3-s + (0.207 + 0.638i)4-s + (−2.01 + 0.972i)5-s + (−3.66 − 1.19i)6-s + 4.10·7-s + (−2.92 − 0.951i)8-s + (−6.62 + 4.81i)9-s + (0.456 − 2.53i)10-s + (1.41 − 1.94i)11-s + (−1.81 + 1.32i)12-s + (0.302 + 0.416i)13-s + (−2.78 + 3.83i)14-s + (−5.17 − 5.39i)15-s + (1.78 − 1.29i)16-s + (4.00 − 0.995i)17-s + ⋯ |
| L(s) = 1 | + (−0.479 + 0.659i)2-s + (0.596 + 1.83i)3-s + (0.103 + 0.319i)4-s + (−0.900 + 0.435i)5-s + (−1.49 − 0.486i)6-s + 1.55·7-s + (−1.03 − 0.336i)8-s + (−2.20 + 1.60i)9-s + (0.144 − 0.802i)10-s + (0.426 − 0.586i)11-s + (−0.524 + 0.381i)12-s + (0.0839 + 0.115i)13-s + (−0.743 + 1.02i)14-s + (−1.33 − 1.39i)15-s + (0.446 − 0.324i)16-s + (0.970 − 0.241i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.924 + 0.380i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.244485 - 1.23684i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.244485 - 1.23684i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.01 - 0.972i)T \) |
| 17 | \( 1 + (-4.00 + 0.995i)T \) |
| good | 2 | \( 1 + (0.677 - 0.932i)T + (-0.618 - 1.90i)T^{2} \) |
| 3 | \( 1 + (-1.03 - 3.18i)T + (-2.42 + 1.76i)T^{2} \) |
| 7 | \( 1 - 4.10T + 7T^{2} \) |
| 11 | \( 1 + (-1.41 + 1.94i)T + (-3.39 - 10.4i)T^{2} \) |
| 13 | \( 1 + (-0.302 - 0.416i)T + (-4.01 + 12.3i)T^{2} \) |
| 19 | \( 1 + (0.992 - 3.05i)T + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + (-0.444 - 0.323i)T + (7.10 + 21.8i)T^{2} \) |
| 29 | \( 1 + (-1.70 + 0.555i)T + (23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-4.70 - 1.52i)T + (25.0 + 18.2i)T^{2} \) |
| 37 | \( 1 + (-7.70 + 5.60i)T + (11.4 - 35.1i)T^{2} \) |
| 41 | \( 1 + (0.481 + 0.662i)T + (-12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 3.79iT - 43T^{2} \) |
| 47 | \( 1 + (7.90 - 2.56i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (5.69 - 1.85i)T + (42.8 - 31.1i)T^{2} \) |
| 59 | \( 1 + (1.99 - 1.44i)T + (18.2 - 56.1i)T^{2} \) |
| 61 | \( 1 + (4.29 - 5.90i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 + (9.58 + 3.11i)T + (54.2 + 39.3i)T^{2} \) |
| 71 | \( 1 + (-13.8 + 4.48i)T + (57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-0.226 - 0.164i)T + (22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (16.0 - 5.21i)T + (63.9 - 46.4i)T^{2} \) |
| 83 | \( 1 + (-0.180 - 0.0585i)T + (67.1 + 48.7i)T^{2} \) |
| 89 | \( 1 + (-0.532 - 0.386i)T + (27.5 + 84.6i)T^{2} \) |
| 97 | \( 1 + (-0.579 - 1.78i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−11.38940464070931598671738116461, −10.77335503077248212987107622477, −9.723382038693031876840747656145, −8.703185767220240263361219929118, −8.176968993797781950074253666318, −7.60452293991142835860203814383, −5.92944954244367409312049391944, −4.69350534136825526691317161318, −3.84252285569008618869709131932, −2.92857451890287653782451377221,
0.977561324220960986529051874135, 1.72809859153554139169919337735, 2.99827486030790191182674530992, 4.79688070476869556550911327457, 6.16575852795871437136952189153, 7.24958524778560805469477390034, 8.151240981529157679408696176697, 8.457625868603554389665156959182, 9.599186299956990108880557392274, 11.15209799071098530774498177351
