| L(s) = 1 | + (−0.412 + 1.35i)2-s + (−0.429 − 2.98i)3-s + (−1.65 − 1.11i)4-s + (−0.959 − 0.281i)5-s + (4.21 + 0.650i)6-s + (−1.41 + 1.63i)7-s + (2.19 − 1.78i)8-s + (−5.85 + 1.71i)9-s + (0.776 − 1.18i)10-s + (0.260 + 0.405i)11-s + (−2.61 + 5.43i)12-s + (−0.398 + 0.345i)13-s + (−1.62 − 2.58i)14-s + (−0.429 + 2.98i)15-s + (1.51 + 3.70i)16-s + (1.00 + 0.457i)17-s + ⋯ |
| L(s) = 1 | + (−0.291 + 0.956i)2-s + (−0.247 − 1.72i)3-s + (−0.829 − 0.557i)4-s + (−0.429 − 0.125i)5-s + (1.72 + 0.265i)6-s + (−0.534 + 0.616i)7-s + (0.775 − 0.631i)8-s + (−1.95 + 0.572i)9-s + (0.245 − 0.373i)10-s + (0.0786 + 0.122i)11-s + (−0.756 + 1.56i)12-s + (−0.110 + 0.0958i)13-s + (−0.434 − 0.691i)14-s + (−0.110 + 0.770i)15-s + (0.377 + 0.925i)16-s + (0.242 + 0.110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.218 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.419604 + 0.336116i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.419604 + 0.336116i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.412 - 1.35i)T \) |
| 5 | \( 1 + (0.959 + 0.281i)T \) |
| 23 | \( 1 + (4.73 - 0.773i)T \) |
| good | 3 | \( 1 + (0.429 + 2.98i)T + (-2.87 + 0.845i)T^{2} \) |
| 7 | \( 1 + (1.41 - 1.63i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.260 - 0.405i)T + (-4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (0.398 - 0.345i)T + (1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.00 - 0.457i)T + (11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (2.30 - 1.05i)T + (12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-7.04 - 3.21i)T + (18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (4.41 + 0.634i)T + (29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-3.42 + 1.00i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (-5.07 - 1.49i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (-9.40 + 1.35i)T + (41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 1.55iT - 47T^{2} \) |
| 53 | \( 1 + (0.419 - 0.483i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (-7.29 - 8.41i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (1.47 - 10.2i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (0.346 - 0.538i)T + (-27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (3.06 - 4.76i)T + (-29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.17 - 11.3i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (-1.22 - 1.41i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (2.13 + 7.27i)T + (-69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (17.3 - 2.49i)T + (85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-4.34 + 14.7i)T + (-81.6 - 52.4i)T^{2} \) |
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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
−10.09675520189611017439114529960, −8.992474911291140603933924952787, −8.357174690999263290954621827919, −7.57167309467610515987013197750, −6.96640850549163287246925625563, −6.08592945493209802504956671509, −5.61508778596623089996307719771, −4.17469449997119171650034889946, −2.50697826156393888577326663721, −1.10918821663830267753443905241,
0.34543084362487466650022571321, 2.67278285411402422523693087941, 3.69844769281094413106309317975, 4.19174947077776840011000525758, 5.06206625870039553925115507888, 6.27457952216051161650843452483, 7.70584260259758696871884757258, 8.585971349542564293511160920170, 9.480212240653047655895378572844, 9.963027711672142272448198459176
