| L(s) = 1 | + (−0.121 − 1.15i)2-s + (−0.978 − 0.207i)3-s + (0.635 − 0.134i)4-s + (−2.64 − 1.92i)5-s + (−0.121 + 1.15i)6-s + (−2.84 + 0.604i)7-s + (−0.951 − 2.92i)8-s + (0.913 + 0.406i)9-s + (−1.89 + 3.28i)10-s + (−1.21 + 3.08i)11-s − 0.649·12-s + (3.14 + 1.75i)13-s + (1.04 + 3.21i)14-s + (2.18 + 2.42i)15-s + (−2.08 + 0.927i)16-s + (0.143 − 1.36i)17-s + ⋯ |
| L(s) = 1 | + (−0.0859 − 0.817i)2-s + (−0.564 − 0.120i)3-s + (0.317 − 0.0674i)4-s + (−1.18 − 0.858i)5-s + (−0.0495 + 0.471i)6-s + (−1.07 + 0.228i)7-s + (−0.336 − 1.03i)8-s + (0.304 + 0.135i)9-s + (−0.600 + 1.04i)10-s + (−0.366 + 0.930i)11-s − 0.187·12-s + (0.873 + 0.487i)13-s + (0.279 + 0.858i)14-s + (0.564 + 0.626i)15-s + (−0.520 + 0.231i)16-s + (0.0349 − 0.332i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.285 - 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0598629 + 0.0803086i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0598629 + 0.0803086i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.978 + 0.207i)T \) |
| 11 | \( 1 + (1.21 - 3.08i)T \) |
| 13 | \( 1 + (-3.14 - 1.75i)T \) |
| good | 2 | \( 1 + (0.121 + 1.15i)T + (-1.95 + 0.415i)T^{2} \) |
| 5 | \( 1 + (2.64 + 1.92i)T + (1.54 + 4.75i)T^{2} \) |
| 7 | \( 1 + (2.84 - 0.604i)T + (6.39 - 2.84i)T^{2} \) |
| 17 | \( 1 + (-0.143 + 1.36i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (3.32 - 3.69i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (1.37 - 2.37i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.99 + 4.43i)T + (-3.03 + 28.8i)T^{2} \) |
| 31 | \( 1 + (2.29 - 1.66i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (1.80 + 2.00i)T + (-3.86 + 36.7i)T^{2} \) |
| 41 | \( 1 + (6.56 + 1.39i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (3.69 + 6.39i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.88 - 8.88i)T + (-38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (-6.88 + 5.00i)T + (16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (9.31 - 1.98i)T + (53.8 - 23.9i)T^{2} \) |
| 61 | \( 1 + (0.928 - 8.83i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-4.92 + 8.53i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-0.829 + 7.89i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (-2.07 + 6.37i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (2.50 - 1.82i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (10.4 + 7.62i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (-0.439 + 0.761i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (6.14 + 2.73i)T + (64.9 + 72.0i)T^{2} \) |
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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
−10.67794294888441859590673509600, −9.819453731920577988635498068501, −8.926338147893555621426340110997, −7.70048037103444043309897508670, −6.77945417133016788270911393122, −5.77115191065874261867693570061, −4.30674509757214913287322114158, −3.48642991126019004073064446501, −1.78654729069123824699234426652, −0.06627493371840430774093941079,
2.97463140033707466417832666940, 3.80778666807804782896215678568, 5.48271540099341824635318306260, 6.48236682923796395631804739427, 6.90336071249069262854790802854, 7.973923718494332458276138536429, 8.727108319187592624664512246430, 10.30951836013029507853874674026, 11.00785645277021077116564311230, 11.47292502940194554593565072197
