| L(s) = 1 | + (0.751 + 1.19i)2-s + (−0.115 − 0.768i)3-s + (−0.869 + 1.80i)4-s + (3.94 + 1.54i)5-s + (0.833 − 0.716i)6-s + (−2.02 + 1.70i)7-s + (−2.81 + 0.313i)8-s + (2.28 − 0.706i)9-s + (1.11 + 5.88i)10-s + (0.256 − 0.829i)11-s + (1.48 + 0.459i)12-s + (−3.90 + 0.892i)13-s + (−3.56 − 1.14i)14-s + (0.732 − 3.21i)15-s + (−2.48 − 3.13i)16-s + (4.54 − 3.09i)17-s + ⋯ |
| L(s) = 1 | + (0.531 + 0.846i)2-s + (−0.0668 − 0.443i)3-s + (−0.434 + 0.900i)4-s + (1.76 + 0.692i)5-s + (0.340 − 0.292i)6-s + (−0.766 + 0.642i)7-s + (−0.993 + 0.110i)8-s + (0.763 − 0.235i)9-s + (0.351 + 1.86i)10-s + (0.0771 − 0.250i)11-s + (0.428 + 0.132i)12-s + (−1.08 + 0.247i)13-s + (−0.951 − 0.307i)14-s + (0.189 − 0.829i)15-s + (−0.622 − 0.782i)16-s + (1.10 − 0.750i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0211 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0211 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44705 + 1.41679i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44705 + 1.41679i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.751 - 1.19i)T \) |
| 7 | \( 1 + (2.02 - 1.70i)T \) |
| good | 3 | \( 1 + (0.115 + 0.768i)T + (-2.86 + 0.884i)T^{2} \) |
| 5 | \( 1 + (-3.94 - 1.54i)T + (3.66 + 3.40i)T^{2} \) |
| 11 | \( 1 + (-0.256 + 0.829i)T + (-9.08 - 6.19i)T^{2} \) |
| 13 | \( 1 + (3.90 - 0.892i)T + (11.7 - 5.64i)T^{2} \) |
| 17 | \( 1 + (-4.54 + 3.09i)T + (6.21 - 15.8i)T^{2} \) |
| 19 | \( 1 + (2.86 - 1.65i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.63 - 1.79i)T + (8.40 + 21.4i)T^{2} \) |
| 29 | \( 1 + (-0.706 + 1.46i)T + (-18.0 - 22.6i)T^{2} \) |
| 31 | \( 1 + (2.09 - 3.62i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (6.13 - 0.459i)T + (36.5 - 5.51i)T^{2} \) |
| 41 | \( 1 + (5.02 + 6.29i)T + (-9.12 + 39.9i)T^{2} \) |
| 43 | \( 1 + (-2.78 - 2.22i)T + (9.56 + 41.9i)T^{2} \) |
| 47 | \( 1 + (-9.02 + 8.37i)T + (3.51 - 46.8i)T^{2} \) |
| 53 | \( 1 + (1.66 + 0.124i)T + (52.4 + 7.89i)T^{2} \) |
| 59 | \( 1 + (-11.8 + 4.63i)T + (43.2 - 40.1i)T^{2} \) |
| 61 | \( 1 + (0.0423 - 0.00317i)T + (60.3 - 9.09i)T^{2} \) |
| 67 | \( 1 + (9.81 + 5.66i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-6.37 + 3.06i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (8.10 + 7.51i)T + (5.45 + 72.7i)T^{2} \) |
| 79 | \( 1 + (-3.55 - 6.14i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (14.1 + 3.22i)T + (74.7 + 36.0i)T^{2} \) |
| 89 | \( 1 + (-0.0985 + 0.0303i)T + (73.5 - 50.1i)T^{2} \) |
| 97 | \( 1 + 2.98T + 97T^{2} \) |
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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
−12.00505309367989537863566448157, −10.28304124478458415275057724081, −9.646011781458590819487257498480, −8.875394000689423154980004184846, −7.23649258881117701089389133220, −6.79174047237643294568683037439, −5.85922907106951690064513912673, −5.17623115230079877022606887887, −3.33142999054634528370217334915, −2.20087643957166395769760504907,
1.34824951724844566143757913632, 2.64028174918018929882009166472, 4.17038488859670098488302106321, 5.07846965686699471739767310639, 5.92026059159427433611185720939, 7.03403790599937256617756589201, 8.869540990920937702821205164624, 9.763953876992225533287212107608, 10.09069978499822193269058404755, 10.69159849003910738661937070452
