| L(s) = 1 | + (−1.73 − 0.779i)2-s + (0.733 + 1.56i)3-s + (1.06 + 1.20i)4-s + (0.850 + 0.666i)5-s + (−0.0473 − 3.28i)6-s + (0.486 − 0.151i)7-s + (0.222 + 0.713i)8-s + (−1.92 + 2.30i)9-s + (−0.953 − 1.81i)10-s + (−5.46 + 2.46i)11-s + (−1.10 + 2.55i)12-s + (2.26 + 2.80i)13-s + (−0.960 − 0.116i)14-s + (−0.421 + 1.82i)15-s + (0.558 − 4.59i)16-s + (−2.94 − 1.54i)17-s + ⋯ |
| L(s) = 1 | + (−1.22 − 0.551i)2-s + (0.423 + 0.905i)3-s + (0.532 + 0.601i)4-s + (0.380 + 0.297i)5-s + (−0.0193 − 1.34i)6-s + (0.183 − 0.0573i)7-s + (0.0786 + 0.252i)8-s + (−0.641 + 0.767i)9-s + (−0.301 − 0.574i)10-s + (−1.64 + 0.741i)11-s + (−0.319 + 0.737i)12-s + (0.626 + 0.779i)13-s + (−0.256 − 0.0311i)14-s + (−0.108 + 0.470i)15-s + (0.139 − 1.14i)16-s + (−0.713 − 0.374i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.535 - 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.272655 + 0.496040i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.272655 + 0.496040i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.733 - 1.56i)T \) |
| 13 | \( 1 + (-2.26 - 2.80i)T \) |
| good | 2 | \( 1 + (1.73 + 0.779i)T + (1.32 + 1.49i)T^{2} \) |
| 5 | \( 1 + (-0.850 - 0.666i)T + (1.19 + 4.85i)T^{2} \) |
| 7 | \( 1 + (-0.486 + 0.151i)T + (5.76 - 3.97i)T^{2} \) |
| 11 | \( 1 + (5.46 - 2.46i)T + (7.29 - 8.23i)T^{2} \) |
| 17 | \( 1 + (2.94 + 1.54i)T + (9.65 + 13.9i)T^{2} \) |
| 19 | \( 1 + (0.193 + 0.193i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.68T + 23T^{2} \) |
| 29 | \( 1 + (4.76 - 1.80i)T + (21.7 - 19.2i)T^{2} \) |
| 31 | \( 1 + (-1.68 - 2.79i)T + (-14.4 + 27.4i)T^{2} \) |
| 37 | \( 1 + (4.77 - 2.88i)T + (17.1 - 32.7i)T^{2} \) |
| 41 | \( 1 + (-0.601 - 3.28i)T + (-38.3 + 14.5i)T^{2} \) |
| 43 | \( 1 + (0.0628 + 0.254i)T + (-38.0 + 19.9i)T^{2} \) |
| 47 | \( 1 + (-0.296 - 4.90i)T + (-46.6 + 5.66i)T^{2} \) |
| 53 | \( 1 + (-2.85 + 5.43i)T + (-30.1 - 43.6i)T^{2} \) |
| 59 | \( 1 + (-0.0837 + 0.106i)T + (-14.1 - 57.2i)T^{2} \) |
| 61 | \( 1 + (-0.717 + 0.376i)T + (34.6 - 50.2i)T^{2} \) |
| 67 | \( 1 + (-0.304 - 5.02i)T + (-66.5 + 8.07i)T^{2} \) |
| 71 | \( 1 + (-2.67 - 14.6i)T + (-66.3 + 25.1i)T^{2} \) |
| 73 | \( 1 + (-2.55 - 5.66i)T + (-48.4 + 54.6i)T^{2} \) |
| 79 | \( 1 + (-8.82 - 7.82i)T + (9.52 + 78.4i)T^{2} \) |
| 83 | \( 1 + (1.36 + 0.250i)T + (77.6 + 29.4i)T^{2} \) |
| 89 | \( 1 + (11.5 + 11.5i)T + 89iT^{2} \) |
| 97 | \( 1 + (7.54 - 5.91i)T + (23.2 - 94.1i)T^{2} \) |
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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
−10.91061701528571500764741626435, −10.16897349493428013152554100077, −9.623862783234256598626101455224, −8.723520923142480235712880100834, −8.069728244530338892204286372813, −7.00052952900323442538390546257, −5.39506682688887280319004620547, −4.51959886175654986527194637914, −2.88437404020639396960954812217, −2.00561658524871009948146171461,
0.45487761830192641132229316820, 1.96785009142482835607597277033, 3.41874547864911016332311351023, 5.40385297708379051173298133499, 6.21169710940859300973423105840, 7.36225286300881736221143448336, 8.010446868998609837982624968535, 8.617515121077418073585879289778, 9.345195622981412693679290272529, 10.50367362150582214178116641794
