| L(s) = 1 | + (1.33 + 0.454i)2-s + (−1.72 − 2.46i)3-s + (1.58 + 1.21i)4-s + (0.318 + 3.64i)5-s + (−1.19 − 4.09i)6-s + (1.89 + 3.27i)7-s + (1.57 + 2.35i)8-s + (−2.07 + 5.70i)9-s + (−1.23 + 5.02i)10-s + (−1.01 − 3.77i)11-s + (0.264 − 6.01i)12-s + (1.58 − 2.26i)13-s + (1.04 + 5.24i)14-s + (8.44 − 7.08i)15-s + (1.03 + 3.86i)16-s + (−1.60 + 0.583i)17-s + ⋯ |
| L(s) = 1 | + (0.946 + 0.321i)2-s + (−0.997 − 1.42i)3-s + (0.793 + 0.608i)4-s + (0.142 + 1.62i)5-s + (−0.486 − 1.66i)6-s + (0.714 + 1.23i)7-s + (0.555 + 0.831i)8-s + (−0.692 + 1.90i)9-s + (−0.389 + 1.58i)10-s + (−0.304 − 1.13i)11-s + (0.0763 − 1.73i)12-s + (0.440 − 0.628i)13-s + (0.278 + 1.40i)14-s + (2.17 − 1.82i)15-s + (0.258 + 0.966i)16-s + (−0.388 + 0.141i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.801 - 0.598i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.69332 + 0.562369i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.69332 + 0.562369i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.33 - 0.454i)T \) |
| 19 | \( 1 + (-3.87 + 2.00i)T \) |
| good | 3 | \( 1 + (1.72 + 2.46i)T + (-1.02 + 2.81i)T^{2} \) |
| 5 | \( 1 + (-0.318 - 3.64i)T + (-4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (-1.89 - 3.27i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.01 + 3.77i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-1.58 + 2.26i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (1.60 - 0.583i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (0.956 - 0.802i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-0.697 - 1.49i)T + (-18.6 + 22.2i)T^{2} \) |
| 31 | \( 1 + (0.632 + 1.09i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-1.77 + 1.77i)T - 37iT^{2} \) |
| 41 | \( 1 + (-1.13 + 6.41i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (2.80 - 0.245i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-3.53 + 9.71i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (0.199 - 2.28i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (13.0 + 6.09i)T + (37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (-0.0525 + 0.600i)T + (-60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (-9.24 + 4.31i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (8.81 - 10.5i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (-6.36 - 1.12i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (2.26 - 12.8i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-2.68 + 10.0i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (0.976 + 5.53i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (2.06 + 5.66i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−11.75532432676877926792987348376, −11.25277995690444692548025241204, −10.70640145437030280475244038421, −8.421978546186656988050080867808, −7.55097249822851453316238369093, −6.70397184758876844033857136497, −5.84360914462294183367894629736, −5.42798585876643770766735718860, −3.12243387504871452154907243793, −2.13279241763509751886136086837,
1.27569319305883984883394184418, 3.95758114827858315880165380299, 4.64115350767422780916832570195, 4.98153532914890005729365419333, 6.18421509415045438132001511900, 7.66841682314572820068816603296, 9.286416454945094018750031088463, 9.982266325653022983142581344380, 10.81724987813263697794898718568, 11.64990112867308282640658461624
