| L(s) = 1 | + (1.29 + 0.574i)2-s + (1.64 + 0.143i)3-s + (1.33 + 1.48i)4-s + (−3.55 + 1.65i)5-s + (2.04 + 1.13i)6-s + (1.19 + 0.688i)7-s + (0.877 + 2.68i)8-s + (−0.272 − 0.0480i)9-s + (−5.54 + 0.0993i)10-s + (3.66 + 0.982i)11-s + (1.98 + 2.63i)12-s + (−0.569 − 6.51i)13-s + (1.14 + 1.57i)14-s + (−6.08 + 2.21i)15-s + (−0.410 + 3.97i)16-s + (0.0136 + 0.0774i)17-s + ⋯ |
| L(s) = 1 | + (0.913 + 0.406i)2-s + (0.949 + 0.0830i)3-s + (0.669 + 0.742i)4-s + (−1.59 + 0.741i)5-s + (0.833 + 0.461i)6-s + (0.450 + 0.260i)7-s + (0.310 + 0.950i)8-s + (−0.0908 − 0.0160i)9-s + (−1.75 + 0.0314i)10-s + (1.10 + 0.296i)11-s + (0.574 + 0.760i)12-s + (−0.158 − 1.80i)13-s + (0.306 + 0.420i)14-s + (−1.57 + 0.571i)15-s + (−0.102 + 0.994i)16-s + (0.00331 + 0.0187i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.353 - 0.935i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.96255 + 1.35683i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.96255 + 1.35683i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.29 - 0.574i)T \) |
| 19 | \( 1 + (-3.84 - 2.05i)T \) |
| good | 3 | \( 1 + (-1.64 - 0.143i)T + (2.95 + 0.520i)T^{2} \) |
| 5 | \( 1 + (3.55 - 1.65i)T + (3.21 - 3.83i)T^{2} \) |
| 7 | \( 1 + (-1.19 - 0.688i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3.66 - 0.982i)T + (9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (0.569 + 6.51i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.0136 - 0.0774i)T + (-15.9 + 5.81i)T^{2} \) |
| 23 | \( 1 + (2.02 + 5.56i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.238 - 0.167i)T + (9.91 - 27.2i)T^{2} \) |
| 31 | \( 1 + (-4.34 + 7.53i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (4.72 - 4.72i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.85 - 2.21i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (2.78 - 1.29i)T + (27.6 - 32.9i)T^{2} \) |
| 47 | \( 1 + (0.0378 - 0.214i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + (4.55 - 9.77i)T + (-34.0 - 40.6i)T^{2} \) |
| 59 | \( 1 + (-0.399 - 0.279i)T + (20.1 + 55.4i)T^{2} \) |
| 61 | \( 1 + (-4.50 - 2.10i)T + (39.2 + 46.7i)T^{2} \) |
| 67 | \( 1 + (1.02 - 0.715i)T + (22.9 - 62.9i)T^{2} \) |
| 71 | \( 1 + (2.71 - 7.46i)T + (-54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (2.26 + 2.69i)T + (-12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (3.11 - 2.61i)T + (13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (7.77 - 2.08i)T + (71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (-7.26 + 8.65i)T + (-15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (3.26 + 18.5i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−11.87372970178092378742382763280, −11.41072250836916175624104692154, −10.16836362304512329223504940871, −8.474640193298600343668578845177, −8.010904039467516752413756368849, −7.25669926838319219804498749049, −5.99026851277026842333686370473, −4.49021819365348304186331011305, −3.51413820626434585390526696031, −2.78581587649342175496687992541,
1.55757148906270255185735246284, 3.40932367944568680755138374145, 4.06438617408014698757910527780, 5.03946483950619517959882095474, 6.80312261073398180408717481283, 7.64657647751350886802240508976, 8.754121770538174201987509120520, 9.433146801856597806053447046685, 11.20787002311183336483360704823, 11.67900677570525398400215724897
