| L(s) = 1 | + (−0.490 + 2.30i)2-s + (2.17 + 2.41i)3-s + (−3.26 − 1.45i)4-s − 1.85i·5-s + (−6.64 + 3.83i)6-s + (−1.05 + 2.35i)7-s + (2.18 − 3.00i)8-s + (−0.789 + 7.51i)9-s + (4.28 + 0.910i)10-s + (0.824 − 0.0866i)11-s + (−3.58 − 11.0i)12-s + (1.84 + 3.09i)13-s + (−4.93 − 3.58i)14-s + (4.47 − 4.03i)15-s + (1.08 + 1.20i)16-s + (0.396 − 3.77i)17-s + ⋯ |
| L(s) = 1 | + (−0.347 + 1.63i)2-s + (1.25 + 1.39i)3-s + (−1.63 − 0.726i)4-s − 0.829i·5-s + (−2.71 + 1.56i)6-s + (−0.396 + 0.891i)7-s + (0.771 − 1.06i)8-s + (−0.263 + 2.50i)9-s + (1.35 + 0.288i)10-s + (0.248 − 0.0261i)11-s + (−1.03 − 3.18i)12-s + (0.512 + 0.858i)13-s + (−1.31 − 0.957i)14-s + (1.15 − 1.04i)15-s + (0.271 + 0.301i)16-s + (0.0962 − 0.915i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.879 + 0.475i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.361652 - 1.42826i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.361652 - 1.42826i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (-1.84 - 3.09i)T \) |
| 31 | \( 1 + (-4.20 + 3.65i)T \) |
| good | 2 | \( 1 + (0.490 - 2.30i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (-2.17 - 2.41i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + 1.85iT - 5T^{2} \) |
| 7 | \( 1 + (1.05 - 2.35i)T + (-4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-0.824 + 0.0866i)T + (10.7 - 2.28i)T^{2} \) |
| 17 | \( 1 + (-0.396 + 3.77i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (3.65 + 3.29i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (-1.70 + 0.757i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (-3.58 - 0.761i)T + (26.4 + 11.7i)T^{2} \) |
| 37 | \( 1 + (7.17 + 4.14i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (2.17 - 10.2i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-0.525 + 0.583i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-4.31 - 1.40i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-7.92 - 5.75i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-1.20 - 5.66i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (-3.52 - 6.10i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (9.72 + 5.61i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.92 + 0.202i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (6.24 + 8.59i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (-0.363 - 0.264i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (7.54 - 2.45i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-11.2 + 1.18i)T + (87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (-7.41 + 16.6i)T + (-64.9 - 72.0i)T^{2} \) |
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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
−11.71617193189488931136243209965, −10.30355615175503314397544178492, −9.257447855208641063917857092803, −8.883852966000244570584740675213, −8.570538545801143691293849960661, −7.32701199453013710113267833079, −6.07771558094297881880616559941, −4.92049372911958431796085572404, −4.36562084195619402120490916823, −2.74002129299427875610279767681,
1.01477098781794859286129517763, 2.18185536911003586063053981872, 3.28659465223978637732511754533, 3.75974328040297138205035656412, 6.38602260479799949486789123170, 7.15620706308667262694989421950, 8.320876979027641690765367051937, 8.791449044358841807058078154515, 10.32131157176252458295353175176, 10.37674262274683651732702699462
