| L(s) = 1 | + (−0.399 + 0.916i)2-s + (1.73 − 0.0282i)3-s + (−0.680 − 0.733i)4-s + (−2.91 − 0.439i)5-s + (−0.666 + 1.59i)6-s + (0.0670 + 0.0621i)7-s + (0.943 − 0.330i)8-s + (2.99 − 0.0977i)9-s + (1.56 − 2.49i)10-s + (2.02 − 1.74i)11-s + (−1.19 − 1.25i)12-s + (3.70 − 5.43i)13-s + (−0.0837 + 0.0365i)14-s + (−5.05 − 0.678i)15-s + (−0.0747 + 0.997i)16-s + (3.35 + 3.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.282 + 0.648i)2-s + (0.999 − 0.0162i)3-s + (−0.340 − 0.366i)4-s + (−1.30 − 0.196i)5-s + (−0.272 + 0.652i)6-s + (0.0253 + 0.0235i)7-s + (0.333 − 0.116i)8-s + (0.999 − 0.0325i)9-s + (0.495 − 0.788i)10-s + (0.611 − 0.526i)11-s + (−0.346 − 0.360i)12-s + (1.02 − 1.50i)13-s + (−0.0223 + 0.00977i)14-s + (−1.30 − 0.175i)15-s + (−0.0186 + 0.249i)16-s + (0.812 + 0.812i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.175i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.49181 + 0.132241i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.49181 + 0.132241i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.399 - 0.916i)T \) |
| 3 | \( 1 + (-1.73 + 0.0282i)T \) |
| 29 | \( 1 + (1.10 - 5.27i)T \) |
| good | 5 | \( 1 + (2.91 + 0.439i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (-0.0670 - 0.0621i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (-2.02 + 1.74i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-3.70 + 5.43i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (-3.35 - 3.35i)T + 17iT^{2} \) |
| 19 | \( 1 + (-2.18 - 1.37i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (0.602 + 0.236i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-4.60 + 3.39i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (-0.513 - 1.46i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (5.81 + 1.55i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (-1.18 + 1.60i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (1.50 + 1.74i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (5.71 + 4.55i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (-0.870 + 0.502i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.181 - 0.00680i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (-8.24 + 0.617i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (4.75 - 2.28i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (0.00563 - 0.0500i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (2.62 - 13.8i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (-2.04 + 6.64i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (-2.45 + 0.276i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (-4.42 - 8.36i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−10.69994594886748560664476834442, −9.862098915389957133890480476486, −8.645487787491598918958161108394, −8.210191889124751419689096705940, −7.67902265957474641268733205091, −6.52420592080385847792335526230, −5.31718575582719953794185113468, −3.85675216576327204634168662872, −3.37017753697151322161430308300, −1.09754277116945214180158863650,
1.43625735544659534368107892758, 2.97286606418916722993292439422, 3.90365075759089055992078937205, 4.54817940414842210550997811315, 6.65712770886203546858574953509, 7.51053988973855004513020685581, 8.232077429535035232558578064866, 9.170083493745627722089692666257, 9.728611295895550048583067821613, 10.96145681786136215317658195199
