| L(s) = 1 | + (1.15 + 0.814i)2-s + (1.84 + 2.63i)3-s + (0.673 + 1.88i)4-s + (−0.158 − 1.81i)5-s + (−0.0128 + 4.54i)6-s + (−1.96 − 3.39i)7-s + (−0.755 + 2.72i)8-s + (−2.51 + 6.89i)9-s + (1.29 − 2.22i)10-s + (0.136 + 0.508i)11-s + (−3.71 + 5.24i)12-s + (0.713 − 1.01i)13-s + (0.499 − 5.52i)14-s + (4.48 − 3.76i)15-s + (−3.09 + 2.53i)16-s + (7.42 − 2.70i)17-s + ⋯ |
| L(s) = 1 | + (0.817 + 0.575i)2-s + (1.06 + 1.52i)3-s + (0.336 + 0.941i)4-s + (−0.0710 − 0.811i)5-s + (−0.00526 + 1.85i)6-s + (−0.741 − 1.28i)7-s + (−0.267 + 0.963i)8-s + (−0.836 + 2.29i)9-s + (0.409 − 0.704i)10-s + (0.0410 + 0.153i)11-s + (−1.07 + 1.51i)12-s + (0.197 − 0.282i)13-s + (0.133 − 1.47i)14-s + (1.15 − 0.972i)15-s + (−0.773 + 0.634i)16-s + (1.80 − 0.655i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.176 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.60621 + 1.91972i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.60621 + 1.91972i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.15 - 0.814i)T \) |
| 19 | \( 1 + (3.99 + 1.74i)T \) |
| good | 3 | \( 1 + (-1.84 - 2.63i)T + (-1.02 + 2.81i)T^{2} \) |
| 5 | \( 1 + (0.158 + 1.81i)T + (-4.92 + 0.868i)T^{2} \) |
| 7 | \( 1 + (1.96 + 3.39i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.136 - 0.508i)T + (-9.52 + 5.5i)T^{2} \) |
| 13 | \( 1 + (-0.713 + 1.01i)T + (-4.44 - 12.2i)T^{2} \) |
| 17 | \( 1 + (-7.42 + 2.70i)T + (13.0 - 10.9i)T^{2} \) |
| 23 | \( 1 + (2.17 - 1.82i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (2.65 + 5.69i)T + (-18.6 + 22.2i)T^{2} \) |
| 31 | \( 1 + (-0.599 - 1.03i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (3.17 - 3.17i)T - 37iT^{2} \) |
| 41 | \( 1 + (-0.564 + 3.20i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-7.30 + 0.639i)T + (42.3 - 7.46i)T^{2} \) |
| 47 | \( 1 + (-0.980 + 2.69i)T + (-36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + (0.861 - 9.85i)T + (-52.1 - 9.20i)T^{2} \) |
| 59 | \( 1 + (11.1 + 5.21i)T + (37.9 + 45.1i)T^{2} \) |
| 61 | \( 1 + (0.272 - 3.10i)T + (-60.0 - 10.5i)T^{2} \) |
| 67 | \( 1 + (2.40 - 1.11i)T + (43.0 - 51.3i)T^{2} \) |
| 71 | \( 1 + (6.04 - 7.20i)T + (-12.3 - 69.9i)T^{2} \) |
| 73 | \( 1 + (0.0541 + 0.00954i)T + (68.5 + 24.9i)T^{2} \) |
| 79 | \( 1 + (1.80 - 10.2i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-0.0522 + 0.195i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 + (1.54 + 8.77i)T + (-83.6 + 30.4i)T^{2} \) |
| 97 | \( 1 + (0.523 + 1.43i)T + (-74.3 + 62.3i)T^{2} \) |
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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
−12.23259447193166276995444198731, −10.82710727141253427529762061726, −9.999398314415996879721432315202, −9.139809734037098972582177355998, −8.140550172418714273528087921176, −7.34773916167233673015495222091, −5.68259382294482813026322306644, −4.57677478663447848604837077156, −3.90495686470403536271295349022, −2.99358284128540876316605910707,
1.76009442699987502356406053524, 2.84615402864966409684399805765, 3.51065184622024792544967989786, 5.87412998399109403477688883019, 6.36946947477238537546219094355, 7.43619574183967467120937485074, 8.600635521587575772870397543481, 9.516344758237228365396988518089, 10.72403999299393477976588646978, 12.04087526287749777063691278591
