| L(s) = 1 | + (2.73 + 0.732i)2-s + (3.73 + 6.46i)3-s + (6.92 + 4i)4-s + (−3.43 + 3.43i)5-s + (5.46 + 20.3i)6-s + (5.09 − 1.36i)7-s + (15.9 + 16i)8-s + (12.6 − 21.9i)9-s + (−11.9 + 6.87i)10-s + (18.2 − 68.0i)11-s + 59.7i·12-s + (−147. − 82.1i)13-s + 14.9·14-s + (−35.0 − 9.38i)15-s + (31.9 + 55.4i)16-s + (−163. − 94.5i)17-s + ⋯ |
| L(s) = 1 | + (0.683 + 0.183i)2-s + (0.414 + 0.718i)3-s + (0.433 + 0.250i)4-s + (−0.137 + 0.137i)5-s + (0.151 + 0.566i)6-s + (0.104 − 0.0278i)7-s + (0.249 + 0.250i)8-s + (0.156 − 0.270i)9-s + (−0.119 + 0.0687i)10-s + (0.150 − 0.562i)11-s + 0.414i·12-s + (−0.873 − 0.486i)13-s + 0.0761·14-s + (−0.155 − 0.0417i)15-s + (0.124 + 0.216i)16-s + (−0.566 − 0.327i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.713 - 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.90328 + 0.777510i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.90328 + 0.777510i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2.73 - 0.732i)T \) |
| 13 | \( 1 + (147. + 82.1i)T \) |
| good | 3 | \( 1 + (-3.73 - 6.46i)T + (-40.5 + 70.1i)T^{2} \) |
| 5 | \( 1 + (3.43 - 3.43i)T - 625iT^{2} \) |
| 7 | \( 1 + (-5.09 + 1.36i)T + (2.07e3 - 1.20e3i)T^{2} \) |
| 11 | \( 1 + (-18.2 + 68.0i)T + (-1.26e4 - 7.32e3i)T^{2} \) |
| 17 | \( 1 + (163. + 94.5i)T + (4.17e4 + 7.23e4i)T^{2} \) |
| 19 | \( 1 + (51.3 + 191. i)T + (-1.12e5 + 6.51e4i)T^{2} \) |
| 23 | \( 1 + (27.6 - 15.9i)T + (1.39e5 - 2.42e5i)T^{2} \) |
| 29 | \( 1 + (-724. - 1.25e3i)T + (-3.53e5 + 6.12e5i)T^{2} \) |
| 31 | \( 1 + (23.2 - 23.2i)T - 9.23e5iT^{2} \) |
| 37 | \( 1 + (203. - 758. i)T + (-1.62e6 - 9.37e5i)T^{2} \) |
| 41 | \( 1 + (439. + 117. i)T + (2.44e6 + 1.41e6i)T^{2} \) |
| 43 | \( 1 + (2.51e3 + 1.45e3i)T + (1.70e6 + 2.96e6i)T^{2} \) |
| 47 | \( 1 + (1.37e3 + 1.37e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 - 3.45e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + (-6.20e3 + 1.66e3i)T + (1.04e7 - 6.05e6i)T^{2} \) |
| 61 | \( 1 + (3.10e3 - 5.38e3i)T + (-6.92e6 - 1.19e7i)T^{2} \) |
| 67 | \( 1 + (2.78e3 + 745. i)T + (1.74e7 + 1.00e7i)T^{2} \) |
| 71 | \( 1 + (-791. - 2.95e3i)T + (-2.20e7 + 1.27e7i)T^{2} \) |
| 73 | \( 1 + (4.76e3 + 4.76e3i)T + 2.83e7iT^{2} \) |
| 79 | \( 1 - 2.82e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + (-7.46e3 + 7.46e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + (-979. + 3.65e3i)T + (-5.43e7 - 3.13e7i)T^{2} \) |
| 97 | \( 1 + (-63.2 - 235. i)T + (-7.66e7 + 4.42e7i)T^{2} \) |
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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
−16.48045627423468707044700866795, −15.35086211030961794288275611256, −14.60380878242848531038501878414, −13.25496612854472803718676387218, −11.80821258113160873964517010102, −10.33986849750646214541612179771, −8.783873625917649147030368291222, −6.94552622341130853083284833195, −4.92955580293316892419119991218, −3.26056089029419311340724104468,
2.14174594879181511453489401984, 4.55143479834713795941466800194, 6.68139429549154780970785242807, 8.088145575041939862798703442861, 10.05751729070934813465558439209, 11.80096834605982323192265366161, 12.80698607498361160185442452159, 13.92043326473997408259952418507, 14.98501440330062831964032861295, 16.40149341605685704625152749174
