| L(s) = 1 | + (0.453 − 0.891i)2-s + (1.72 − 0.102i)3-s + (−0.587 − 0.809i)4-s + (1.55 − 1.60i)5-s + (0.693 − 1.58i)6-s + (−2.97 + 2.97i)7-s + (−0.987 + 0.156i)8-s + (2.97 − 0.353i)9-s + (−0.729 − 2.11i)10-s + (−4.73 + 1.53i)11-s + (−1.09 − 1.33i)12-s + (1.57 − 0.801i)13-s + (1.30 + 4.00i)14-s + (2.51 − 2.94i)15-s + (−0.309 + 0.951i)16-s + (0.223 + 1.40i)17-s + ⋯ |
| L(s) = 1 | + (0.321 − 0.630i)2-s + (0.998 − 0.0589i)3-s + (−0.293 − 0.404i)4-s + (0.694 − 0.719i)5-s + (0.283 − 0.647i)6-s + (−1.12 + 1.12i)7-s + (−0.349 + 0.0553i)8-s + (0.993 − 0.117i)9-s + (−0.230 − 0.668i)10-s + (−1.42 + 0.464i)11-s + (−0.317 − 0.386i)12-s + (0.436 − 0.222i)13-s + (0.347 + 1.07i)14-s + (0.650 − 0.759i)15-s + (−0.0772 + 0.237i)16-s + (0.0541 + 0.341i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.45083 - 0.738196i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.45083 - 0.738196i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.453 + 0.891i)T \) |
| 3 | \( 1 + (-1.72 + 0.102i)T \) |
| 5 | \( 1 + (-1.55 + 1.60i)T \) |
| good | 7 | \( 1 + (2.97 - 2.97i)T - 7iT^{2} \) |
| 11 | \( 1 + (4.73 - 1.53i)T + (8.89 - 6.46i)T^{2} \) |
| 13 | \( 1 + (-1.57 + 0.801i)T + (7.64 - 10.5i)T^{2} \) |
| 17 | \( 1 + (-0.223 - 1.40i)T + (-16.1 + 5.25i)T^{2} \) |
| 19 | \( 1 + (-1.09 + 1.50i)T + (-5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.86 - 0.951i)T + (13.5 + 18.6i)T^{2} \) |
| 29 | \( 1 + (4.57 - 3.32i)T + (8.96 - 27.5i)T^{2} \) |
| 31 | \( 1 + (5.23 + 3.80i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.16 - 2.29i)T + (-21.7 + 29.9i)T^{2} \) |
| 41 | \( 1 + (-3.58 - 1.16i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (-6.26 - 6.26i)T + 43iT^{2} \) |
| 47 | \( 1 + (4.62 + 0.732i)T + (44.6 + 14.5i)T^{2} \) |
| 53 | \( 1 + (0.801 - 5.05i)T + (-50.4 - 16.3i)T^{2} \) |
| 59 | \( 1 + (-3.44 + 10.5i)T + (-47.7 - 34.6i)T^{2} \) |
| 61 | \( 1 + (2.99 + 9.23i)T + (-49.3 + 35.8i)T^{2} \) |
| 67 | \( 1 + (6.39 - 1.01i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (8.85 + 12.1i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-4.19 + 8.23i)T + (-42.9 - 59.0i)T^{2} \) |
| 79 | \( 1 + (-2.20 - 3.03i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-16.6 + 2.63i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (0.351 + 1.08i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (-0.549 + 3.47i)T + (-92.2 - 29.9i)T^{2} \) |
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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
−13.00537268055387355973816164415, −12.39076076379919037926872232510, −10.67607079648082167949313636624, −9.525845530189653594572239807544, −9.157735409713852324129316363057, −7.87428355722194160573346467317, −6.11020744947167184755280054617, −4.99212480803931483252883253105, −3.22177654926819406552029967192, −2.14738974236874811875168652891,
2.81948540917366161994278770206, 3.84683575749951370167324162218, 5.67213192953307214145592215057, 6.96877712208668539087588215518, 7.61923770962313608661633003847, 9.043799922840303216113132507163, 10.04036633183392721548305580457, 10.78312371621055032262923164018, 12.87854335649740491755366160569, 13.40316239066542086423179909937
