| L(s) = 1 | − 1.31i·2-s + 2.27·4-s − 1.27·5-s + 4.72·7-s − 8.24i·8-s + 1.67i·10-s + 7.27·11-s − 4.30i·13-s − 6.20i·14-s − 1.72·16-s + 20.3·17-s + (−7.54 − 17.4i)19-s − 2.90·20-s − 9.55i·22-s − 5.45·23-s + ⋯ |
| L(s) = 1 | − 0.656i·2-s + 0.568·4-s − 0.254·5-s + 0.675·7-s − 1.03i·8-s + 0.167i·10-s + 0.661·11-s − 0.330i·13-s − 0.443i·14-s − 0.107·16-s + 1.19·17-s + (−0.397 − 0.917i)19-s − 0.145·20-s − 0.434i·22-s − 0.236·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 171 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.397 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.51553 - 0.995272i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.51553 - 0.995272i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 19 | \( 1 + (7.54 + 17.4i)T \) |
| good | 2 | \( 1 + 1.31iT - 4T^{2} \) |
| 5 | \( 1 + 1.27T + 25T^{2} \) |
| 7 | \( 1 - 4.72T + 49T^{2} \) |
| 11 | \( 1 - 7.27T + 121T^{2} \) |
| 13 | \( 1 + 4.30iT - 169T^{2} \) |
| 17 | \( 1 - 20.3T + 289T^{2} \) |
| 23 | \( 1 + 5.45T + 529T^{2} \) |
| 29 | \( 1 + 8.60iT - 841T^{2} \) |
| 31 | \( 1 - 20.0iT - 961T^{2} \) |
| 37 | \( 1 - 40.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 - 31.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 65.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 55.4T + 2.20e3T^{2} \) |
| 53 | \( 1 - 78.1iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 69.5iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 6.17T + 3.72e3T^{2} \) |
| 67 | \( 1 - 123. iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 3.11iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 33.8T + 5.32e3T^{2} \) |
| 79 | \( 1 + 87.6iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 121.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 69.9iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 111. iT - 9.40e3T^{2} \) |
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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
−12.02409561222097370453706550256, −11.49617053472485721710172228820, −10.52812049619537081963190581027, −9.560727669931289425333523989883, −8.181136492830774776003697475921, −7.17801290218385214845293732685, −5.92493652552956342868559063646, −4.32499183034106497311626165930, −2.95902826947049590933307668204, −1.34440747902414432283504878228,
1.86243604222332437450711433873, 3.80462921533022086315625223545, 5.37594650658640332027900972082, 6.38666782000813674260058075019, 7.59925175451755589647156947977, 8.227330660187433701326728089669, 9.626059349216559874165574805213, 10.90454451548893832829227714888, 11.70127309809162430757959425952, 12.50569603772741034652808090307
