| L(s) = 1 | + (−0.946 − 1.05i)2-s + (0.977 − 1.42i)3-s + (−0.208 + 1.98i)4-s + (0.0906 + 0.248i)5-s + (−2.42 + 0.326i)6-s + (−0.984 + 0.173i)7-s + (2.28 − 1.66i)8-s + (−1.08 − 2.79i)9-s + (0.175 − 0.330i)10-s + (5.73 + 2.08i)11-s + (2.64 + 2.24i)12-s + (1.01 + 0.851i)13-s + (1.11 + 0.870i)14-s + (0.444 + 0.113i)15-s + (−3.91 − 0.829i)16-s + (2.19 + 1.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.669 − 0.743i)2-s + (0.564 − 0.825i)3-s + (−0.104 + 0.994i)4-s + (0.0405 + 0.111i)5-s + (−0.991 + 0.133i)6-s + (−0.372 + 0.0656i)7-s + (0.808 − 0.588i)8-s + (−0.363 − 0.931i)9-s + (0.0556 − 0.104i)10-s + (1.72 + 0.628i)11-s + (0.762 + 0.647i)12-s + (0.281 + 0.236i)13-s + (0.297 + 0.232i)14-s + (0.114 + 0.0293i)15-s + (−0.978 − 0.207i)16-s + (0.532 + 0.307i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0747 + 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0747 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.03212 - 0.957668i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.03212 - 0.957668i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.946 + 1.05i)T \) |
| 3 | \( 1 + (-0.977 + 1.42i)T \) |
| 7 | \( 1 + (0.984 - 0.173i)T \) |
| good | 5 | \( 1 + (-0.0906 - 0.248i)T + (-3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-5.73 - 2.08i)T + (8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-1.01 - 0.851i)T + (2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-2.19 - 1.26i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.96 + 1.70i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.696 - 3.94i)T + (-21.6 - 7.86i)T^{2} \) |
| 29 | \( 1 + (2.88 + 3.43i)T + (-5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (2.67 + 0.470i)T + (29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (-3.66 + 6.34i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.34 + 1.59i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (-1.74 + 4.78i)T + (-32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-1.79 - 10.2i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + 7.06iT - 53T^{2} \) |
| 59 | \( 1 + (-1.92 + 0.702i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (1.40 + 7.98i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (3.41 - 4.07i)T + (-11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (3.68 - 6.38i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.77 - 6.54i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (6.83 + 8.14i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (2.06 - 1.73i)T + (14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-7.43 + 4.29i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (16.7 + 6.10i)T + (74.3 + 62.3i)T^{2} \) |
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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
−9.805681709995177229155337237223, −9.330043557879134690211395804760, −8.661169870509738664362189672220, −7.55707294698690545901233284838, −6.98834226566806721682271102356, −6.02474703769876008190936217207, −4.13639597708232466228606526339, −3.37134081768197372334400459079, −2.12988622685232584265670272755, −1.07698588230094140089861163293,
1.27959536657730048549604167773, 3.13716057391478922689865878356, 4.17127595533379673431922460683, 5.31835351745860447526370552166, 6.20068383739035400307868439699, 7.17614336424578547300140880330, 8.172350641885234192997762461738, 9.013087421775944529344889537871, 9.373033345890673915990345635255, 10.27529000441828642445559752012
