| L(s) = 1 | + (2.11 + 0.110i)2-s + (1.20 + 1.48i)3-s + (2.46 + 0.259i)4-s + (−2.22 + 0.177i)5-s + (2.38 + 3.28i)6-s + (−1.95 − 1.78i)7-s + (0.998 + 0.158i)8-s + (−0.139 + 0.657i)9-s + (−4.72 + 0.129i)10-s + (4.09 − 0.870i)11-s + (2.58 + 3.98i)12-s + (−1.02 − 0.521i)13-s + (−3.92 − 3.98i)14-s + (−2.95 − 3.10i)15-s + (−2.75 − 0.585i)16-s + (−0.877 − 2.28i)17-s + ⋯ |
| L(s) = 1 | + (1.49 + 0.0783i)2-s + (0.696 + 0.859i)3-s + (1.23 + 0.129i)4-s + (−0.996 + 0.0795i)5-s + (0.973 + 1.33i)6-s + (−0.738 − 0.674i)7-s + (0.353 + 0.0559i)8-s + (−0.0465 + 0.219i)9-s + (−1.49 + 0.0408i)10-s + (1.23 − 0.262i)11-s + (0.746 + 1.14i)12-s + (−0.283 − 0.144i)13-s + (−1.05 − 1.06i)14-s + (−0.762 − 0.801i)15-s + (−0.688 − 0.146i)16-s + (−0.212 − 0.554i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.27984 + 0.693572i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.27984 + 0.693572i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (2.22 - 0.177i)T \) |
| 7 | \( 1 + (1.95 + 1.78i)T \) |
| good | 2 | \( 1 + (-2.11 - 0.110i)T + (1.98 + 0.209i)T^{2} \) |
| 3 | \( 1 + (-1.20 - 1.48i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (-4.09 + 0.870i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (1.02 + 0.521i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (0.877 + 2.28i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.639 - 6.08i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.0765 - 1.46i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (4.09 - 5.63i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (1.60 - 3.60i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (-7.90 + 5.13i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (6.68 - 2.17i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (3.56 + 3.56i)T + 43iT^{2} \) |
| 47 | \( 1 + (1.97 + 0.756i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (7.54 - 6.10i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-3.93 + 4.36i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-2.12 + 1.90i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-8.82 + 3.38i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (2.38 + 1.73i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-1.30 + 2.00i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (2.05 + 4.61i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (-0.592 + 3.74i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (-10.1 - 11.2i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (1.77 + 11.2i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−12.84621023074532313955831231380, −12.08342893909125336569019860543, −11.11519569982867294420810758357, −9.818002297432705530376733936246, −8.860543218838663512937955985771, −7.34912167891262277035218400058, −6.32883193663396619849752230132, −4.73686546891248353946880095982, −3.62876813639912762215536641939, −3.44246371987260107948915640267,
2.41132265865844072942134324927, 3.59150449142716188224128382095, 4.72380614112378309676868901113, 6.34145994664086396514519173230, 7.06937593298322235127988482704, 8.393871498301416609520377910031, 9.397397678886844560179195840400, 11.40689460416534092777554137689, 11.91049101823773698837495880998, 12.96520296933390464594031976012
