| L(s) = 1 | + (0.834 − 1.62i)2-s + (−0.555 − 5.05i)3-s + (0.389 + 0.544i)4-s + (6.62 + 3.10i)5-s + (−8.66 − 3.31i)6-s + (1.68 + 7.57i)7-s + (8.43 − 1.24i)8-s + (−16.4 + 3.65i)9-s + (10.5 − 8.15i)10-s + (4.07 − 2.28i)11-s + (2.53 − 2.27i)12-s + (−3.71 + 1.90i)13-s + (13.7 + 3.58i)14-s + (12.0 − 35.1i)15-s + (4.15 − 12.1i)16-s + (−10.2 − 18.3i)17-s + ⋯ |
| L(s) = 1 | + (0.417 − 0.811i)2-s + (−0.185 − 1.68i)3-s + (0.0974 + 0.136i)4-s + (1.32 + 0.621i)5-s + (−1.44 − 0.552i)6-s + (0.241 + 1.08i)7-s + (1.05 − 0.155i)8-s + (−1.82 + 0.406i)9-s + (1.05 − 0.815i)10-s + (0.370 − 0.208i)11-s + (0.211 − 0.189i)12-s + (−0.285 + 0.146i)13-s + (0.979 + 0.256i)14-s + (0.800 − 2.34i)15-s + (0.259 − 0.761i)16-s + (−0.605 − 1.07i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 431 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.202 + 0.979i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.83490 - 2.25344i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.83490 - 2.25344i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 431 | \( 1 + (429. + 38.3i)T \) |
| good | 2 | \( 1 + (-0.834 + 1.62i)T + (-2.32 - 3.25i)T^{2} \) |
| 3 | \( 1 + (0.555 + 5.05i)T + (-8.78 + 1.95i)T^{2} \) |
| 5 | \( 1 + (-6.62 - 3.10i)T + (15.9 + 19.2i)T^{2} \) |
| 7 | \( 1 + (-1.68 - 7.57i)T + (-44.3 + 20.7i)T^{2} \) |
| 11 | \( 1 + (-4.07 + 2.28i)T + (63.0 - 103. i)T^{2} \) |
| 13 | \( 1 + (3.71 - 1.90i)T + (98.3 - 137. i)T^{2} \) |
| 17 | \( 1 + (10.2 + 18.3i)T + (-150. + 246. i)T^{2} \) |
| 19 | \( 1 + (-26.8 + 12.5i)T + (230. - 277. i)T^{2} \) |
| 23 | \( 1 + (1.65 + 1.71i)T + (-19.3 + 528. i)T^{2} \) |
| 29 | \( 1 + (-6.96 + 9.73i)T + (-271. - 795. i)T^{2} \) |
| 31 | \( 1 + (5.81 + 15.1i)T + (-715. + 641. i)T^{2} \) |
| 37 | \( 1 + (16.6 + 4.36i)T + (1.19e3 + 670. i)T^{2} \) |
| 41 | \( 1 + (9.02 - 26.4i)T + (-1.33e3 - 1.02e3i)T^{2} \) |
| 43 | \( 1 + (11.9 - 15.4i)T + (-467. - 1.78e3i)T^{2} \) |
| 47 | \( 1 + (4.26 - 58.2i)T + (-2.18e3 - 321. i)T^{2} \) |
| 53 | \( 1 + (-45.6 + 3.33i)T + (2.77e3 - 408. i)T^{2} \) |
| 59 | \( 1 + (23.4 - 45.6i)T + (-2.02e3 - 2.83e3i)T^{2} \) |
| 61 | \( 1 + (42.7 + 100. i)T + (-2.58e3 + 2.67e3i)T^{2} \) |
| 67 | \( 1 + (102. + 26.8i)T + (3.91e3 + 2.19e3i)T^{2} \) |
| 71 | \( 1 + (8.46 - 115. i)T + (-4.98e3 - 733. i)T^{2} \) |
| 73 | \( 1 + (-68.0 - 102. i)T + (-2.08e3 + 4.90e3i)T^{2} \) |
| 79 | \( 1 + (-0.0698 + 0.232i)T + (-5.20e3 - 3.44e3i)T^{2} \) |
| 83 | \( 1 + (61.2 - 6.73i)T + (6.72e3 - 1.49e3i)T^{2} \) |
| 89 | \( 1 + (48.4 - 8.95i)T + (7.39e3 - 2.82e3i)T^{2} \) |
| 97 | \( 1 + (-42.0 + 12.6i)T + (7.84e3 - 5.19e3i)T^{2} \) |
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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
−11.31162445908612108978269784039, −9.877872035267367662087041050008, −8.938943309295486892860562217828, −7.64578771451439856780742070732, −6.88441287815151932080996932157, −6.05948536553145036947774555152, −5.08175980094882846412363444027, −2.82165042885668209236122151652, −2.39905451297708194830211085888, −1.35892269442789498086938730437,
1.56537090863694857887723635179, 3.72161178495491579433424815731, 4.69549536126449970307503666084, 5.34790858309710404545775134973, 6.11133561502563668333538864617, 7.30030093822099714110119102155, 8.675195508182321560236011677896, 9.588431767533309353676514376469, 10.43243539175299955314290584816, 10.55936861147465051880249087484
