| L(s) = 1 | + (1.25 − 1.70i)2-s + (−1.93 + 0.365i)3-s + (−0.728 − 2.36i)4-s + (−1.72 − 1.41i)5-s + (−1.80 + 3.75i)6-s + (−2.39 − 1.11i)7-s + (−0.944 − 0.330i)8-s + (0.813 − 0.319i)9-s + (−4.58 + 1.16i)10-s + (0.0928 − 0.236i)11-s + (2.27 + 4.30i)12-s + (−2.22 − 0.251i)13-s + (−4.90 + 2.68i)14-s + (3.86 + 2.10i)15-s + (2.33 − 1.59i)16-s + (−3.82 − 0.143i)17-s + ⋯ |
| L(s) = 1 | + (0.888 − 1.20i)2-s + (−1.11 + 0.211i)3-s + (−0.364 − 1.18i)4-s + (−0.773 − 0.633i)5-s + (−0.737 + 1.53i)6-s + (−0.907 − 0.421i)7-s + (−0.333 − 0.116i)8-s + (0.271 − 0.106i)9-s + (−1.44 + 0.367i)10-s + (0.0279 − 0.0713i)11-s + (0.656 + 1.24i)12-s + (−0.618 − 0.0696i)13-s + (−1.31 + 0.717i)14-s + (0.997 + 0.544i)15-s + (0.584 − 0.398i)16-s + (−0.928 − 0.0347i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.0956785 + 0.729612i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0956785 + 0.729612i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + (1.72 + 1.41i)T \) |
| 7 | \( 1 + (2.39 + 1.11i)T \) |
| good | 2 | \( 1 + (-1.25 + 1.70i)T + (-0.589 - 1.91i)T^{2} \) |
| 3 | \( 1 + (1.93 - 0.365i)T + (2.79 - 1.09i)T^{2} \) |
| 11 | \( 1 + (-0.0928 + 0.236i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (2.22 + 0.251i)T + (12.6 + 2.89i)T^{2} \) |
| 17 | \( 1 + (3.82 + 0.143i)T + (16.9 + 1.27i)T^{2} \) |
| 19 | \( 1 + (-1.87 + 3.24i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.169 + 4.52i)T + (-22.9 + 1.71i)T^{2} \) |
| 29 | \( 1 + (0.249 - 0.0570i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (-6.33 + 3.65i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.555 + 0.293i)T + (20.8 - 30.5i)T^{2} \) |
| 41 | \( 1 + (3.39 + 7.04i)T + (-25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-0.0831 - 0.237i)T + (-33.6 + 26.8i)T^{2} \) |
| 47 | \( 1 + (-7.95 - 5.86i)T + (13.8 + 44.9i)T^{2} \) |
| 53 | \( 1 + (9.66 + 5.10i)T + (29.8 + 43.7i)T^{2} \) |
| 59 | \( 1 + (-0.928 - 12.3i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (2.88 - 9.36i)T + (-50.4 - 34.3i)T^{2} \) |
| 67 | \( 1 + (11.0 + 2.95i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (-2.31 + 10.1i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.123 + 0.0908i)T + (21.5 - 69.7i)T^{2} \) |
| 79 | \( 1 + (0.348 + 0.201i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (1.84 + 16.3i)T + (-80.9 + 18.4i)T^{2} \) |
| 89 | \( 1 + (0.234 + 0.596i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + (0.348 + 0.348i)T + 97iT^{2} \) |
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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
−11.76336207432018013201675667911, −10.90546890634357159251715066847, −10.21319630033405978052988249482, −8.988776219052254029589631053325, −7.39960405407656094868478353170, −6.09625816786844804848179213968, −4.84223108643899716252811688960, −4.26149078662999029814420685683, −2.84214384870822108155021680571, −0.49789470778302437400067192702,
3.27197446715708715020361457056, 4.61089572717401879621513819297, 5.69934135359570443413994156676, 6.52530930330511818210804926605, 7.09671038588710913947335596493, 8.244611051705761067690690930634, 9.829434720609844591053056400834, 10.97818181260307433950740841515, 11.98057130134429070728617618647, 12.51708368870981069848471145357
