L(s) = 1 | + (2.23 − 0.103i)5-s + (−1.10 + 0.640i)7-s + (2.07 + 3.58i)11-s + (5.64 + 3.26i)13-s + 5.98i·17-s − 7.17·19-s + (−6.52 − 3.76i)23-s + (4.97 − 0.461i)25-s + (−2.59 − 4.5i)29-s + (−2.58 + 4.48i)31-s + (−2.41 + 1.54i)35-s + 5.24i·37-s + (0.340 − 0.589i)41-s + (1.10 − 0.640i)43-s + (4.59 − 2.65i)47-s + ⋯ |
L(s) = 1 | + (0.998 − 0.0462i)5-s + (−0.419 + 0.242i)7-s + (0.624 + 1.08i)11-s + (1.56 + 0.904i)13-s + 1.45i·17-s − 1.64·19-s + (−1.35 − 0.785i)23-s + (0.995 − 0.0923i)25-s + (−0.482 − 0.835i)29-s + (−0.465 + 0.805i)31-s + (−0.407 + 0.261i)35-s + 0.861i·37-s + (0.0531 − 0.0920i)41-s + (0.169 − 0.0977i)43-s + (0.670 − 0.387i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.298 - 0.954i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.886002042\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.886002042\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.23 + 0.103i)T \) |
good | 7 | \( 1 + (1.10 - 0.640i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.07 - 3.58i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.64 - 3.26i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 - 5.98iT - 17T^{2} \) |
| 19 | \( 1 + 7.17T + 19T^{2} \) |
| 23 | \( 1 + (6.52 + 3.76i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.59 + 4.5i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.58 - 4.48i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.24iT - 37T^{2} \) |
| 41 | \( 1 + (-0.340 + 0.589i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.10 + 0.640i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.59 + 2.65i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 2.21iT - 53T^{2} \) |
| 59 | \( 1 + (-3.80 + 6.58i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.08 - 1.88i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-13.5 - 7.80i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 5.50T + 71T^{2} \) |
| 73 | \( 1 - 7.80iT - 73T^{2} \) |
| 79 | \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-8.44 + 4.87i)T + (41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 10.0T + 89T^{2} \) |
| 97 | \( 1 + (-12.4 + 7.16i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.573511457841718287508163180119, −8.725761120163765051896065074816, −8.293827705831403080743583623965, −6.72740557079575702773365106444, −6.39909692165787782041483116080, −5.75319132288522652880803185131, −4.34497524976520158018753995564, −3.84231218891319328634521973959, −2.19362699795057648520404261815, −1.62787322620764174368517066621,
0.72689406173775826664261221204, 2.04386572690435973831271404783, 3.26357669605477319466377422091, 3.98469989422692200597202715772, 5.40680378348403521235371490167, 6.02326837515154267448521340552, 6.55311458693297112147428434050, 7.67579178151615547681576869094, 8.650649918408773397451673578220, 9.164405228794698668490751448476