| L(s) = 1 | + (−0.346 − 0.129i)3-s + (2.23 − 0.0500i)5-s + (2.73 − 0.595i)7-s + (−2.16 − 1.87i)9-s + (1.78 − 0.256i)11-s + (−1.03 + 4.77i)13-s + (−0.780 − 0.271i)15-s + (2.53 − 4.64i)17-s + (−1.01 − 0.297i)19-s + (−1.02 − 0.147i)21-s + (−1.23 − 4.63i)23-s + (4.99 − 0.223i)25-s + (1.03 + 1.90i)27-s + (2.32 + 7.90i)29-s + (1.26 − 2.77i)31-s + ⋯ |
| L(s) = 1 | + (−0.199 − 0.0745i)3-s + (0.999 − 0.0223i)5-s + (1.03 − 0.225i)7-s + (−0.721 − 0.625i)9-s + (0.537 − 0.0773i)11-s + (−0.288 + 1.32i)13-s + (−0.201 − 0.0700i)15-s + (0.615 − 1.12i)17-s + (−0.232 − 0.0681i)19-s + (−0.223 − 0.0321i)21-s + (−0.257 − 0.966i)23-s + (0.998 − 0.0447i)25-s + (0.199 + 0.366i)27-s + (0.431 + 1.46i)29-s + (0.227 − 0.497i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.314i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.64930 - 0.265931i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.64930 - 0.265931i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.23 + 0.0500i)T \) |
| 23 | \( 1 + (1.23 + 4.63i)T \) |
| good | 3 | \( 1 + (0.346 + 0.129i)T + (2.26 + 1.96i)T^{2} \) |
| 7 | \( 1 + (-2.73 + 0.595i)T + (6.36 - 2.90i)T^{2} \) |
| 11 | \( 1 + (-1.78 + 0.256i)T + (10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (1.03 - 4.77i)T + (-11.8 - 5.40i)T^{2} \) |
| 17 | \( 1 + (-2.53 + 4.64i)T + (-9.19 - 14.3i)T^{2} \) |
| 19 | \( 1 + (1.01 + 0.297i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.32 - 7.90i)T + (-24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-1.26 + 2.77i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (-0.259 - 3.62i)T + (-36.6 + 5.26i)T^{2} \) |
| 41 | \( 1 + (-6.73 - 7.77i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.20 + 11.2i)T + (-32.4 - 28.1i)T^{2} \) |
| 47 | \( 1 + (4.87 + 4.87i)T + 47iT^{2} \) |
| 53 | \( 1 + (-0.222 - 1.02i)T + (-48.2 + 22.0i)T^{2} \) |
| 59 | \( 1 + (0.486 + 0.756i)T + (-24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (7.03 + 3.21i)T + (39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (2.21 + 2.96i)T + (-18.8 + 64.2i)T^{2} \) |
| 71 | \( 1 + (2.00 - 13.9i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (9.51 - 5.19i)T + (39.4 - 61.4i)T^{2} \) |
| 79 | \( 1 + (11.7 - 7.57i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (2.76 - 0.197i)T + (82.1 - 11.8i)T^{2} \) |
| 89 | \( 1 + (4.59 + 10.0i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (6.02 + 0.430i)T + (96.0 + 13.8i)T^{2} \) |
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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
−11.17112986883117036666692081690, −10.06356321981693981521625389071, −9.169817049944871327853060790177, −8.533434597480318827913027660950, −7.12957406032725159305503328795, −6.37090549025466658139228684506, −5.30644349529323348057591706098, −4.38072256832255692987083483059, −2.71834382995530138763598253719, −1.33995789873750758235036227997,
1.58349358467002012930273998300, 2.83800264345050916125400591307, 4.51626899566085380459788871154, 5.63538581244318087320347306354, 6.00395545687129910170670337129, 7.69532902860438909465206463205, 8.274875551214554289344448014483, 9.375985952555633178765753777794, 10.35195666961844248199121751086, 10.94655651100082409624916409563
