| L(s) = 1 | + (0.998 + 0.0523i)2-s + (0.449 + 0.554i)3-s + (0.994 + 0.104i)4-s + (−0.415 − 2.19i)5-s + (0.419 + 0.577i)6-s + (1.35 − 2.27i)7-s + (0.987 + 0.156i)8-s + (0.517 − 2.43i)9-s + (−0.299 − 2.21i)10-s + (−1.05 + 0.224i)11-s + (0.388 + 0.598i)12-s + (−2.33 − 1.18i)13-s + (1.47 − 2.19i)14-s + (1.03 − 1.21i)15-s + (0.978 + 0.207i)16-s + (2.67 + 6.97i)17-s + ⋯ |
| L(s) = 1 | + (0.706 + 0.0370i)2-s + (0.259 + 0.320i)3-s + (0.497 + 0.0522i)4-s + (−0.185 − 0.982i)5-s + (0.171 + 0.235i)6-s + (0.513 − 0.858i)7-s + (0.349 + 0.0553i)8-s + (0.172 − 0.811i)9-s + (−0.0947 − 0.700i)10-s + (−0.319 + 0.0678i)11-s + (0.112 + 0.172i)12-s + (−0.646 − 0.329i)13-s + (0.394 − 0.587i)14-s + (0.266 − 0.314i)15-s + (0.244 + 0.0519i)16-s + (0.648 + 1.69i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.871 + 0.489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.09357 - 0.548002i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.09357 - 0.548002i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.998 - 0.0523i)T \) |
| 5 | \( 1 + (0.415 + 2.19i)T \) |
| 7 | \( 1 + (-1.35 + 2.27i)T \) |
| good | 3 | \( 1 + (-0.449 - 0.554i)T + (-0.623 + 2.93i)T^{2} \) |
| 11 | \( 1 + (1.05 - 0.224i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (2.33 + 1.18i)T + (7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (-2.67 - 6.97i)T + (-12.6 + 11.3i)T^{2} \) |
| 19 | \( 1 + (-0.683 - 6.50i)T + (-18.5 + 3.95i)T^{2} \) |
| 23 | \( 1 + (0.00179 - 0.0341i)T + (-22.8 - 2.40i)T^{2} \) |
| 29 | \( 1 + (-4.62 + 6.37i)T + (-8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (2.20 - 4.94i)T + (-20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (2.08 - 1.35i)T + (15.0 - 33.8i)T^{2} \) |
| 41 | \( 1 + (7.29 - 2.36i)T + (33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-8.94 - 8.94i)T + 43iT^{2} \) |
| 47 | \( 1 + (7.97 + 3.06i)T + (34.9 + 31.4i)T^{2} \) |
| 53 | \( 1 + (1.35 - 1.09i)T + (11.0 - 51.8i)T^{2} \) |
| 59 | \( 1 + (-8.09 + 8.99i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.724 + 0.652i)T + (6.37 - 60.6i)T^{2} \) |
| 67 | \( 1 + (-7.37 + 2.82i)T + (49.7 - 44.8i)T^{2} \) |
| 71 | \( 1 + (5.21 + 3.78i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (3.45 - 5.32i)T + (-29.6 - 66.6i)T^{2} \) |
| 79 | \( 1 + (0.297 + 0.668i)T + (-52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (0.974 - 6.15i)T + (-78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (8.61 + 9.57i)T + (-9.30 + 88.5i)T^{2} \) |
| 97 | \( 1 + (-2.66 - 16.8i)T + (-92.2 + 29.9i)T^{2} \) |
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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
−11.67824738712189087117987641731, −10.34710203772734327721191931689, −9.835739561279259388879126429411, −8.306059377242555001796831662786, −7.85250220627566043054571864349, −6.42527675068838664087517674578, −5.29120484956296306566884131463, −4.26522905789728274716799218030, −3.52197433949675360165249853563, −1.44088631658273952017457609393,
2.30094375615831364743994095109, 2.95819138205187035276918693142, 4.74153954963894617334723714982, 5.47667467496319648480158450343, 7.03853636961427643264019625361, 7.37979381392601167935514218470, 8.648169238575190054220637992577, 9.863660963426877799453337588097, 10.95590152684710131923763155404, 11.59035071805579417820838707262
