L(s) = 1 | + (−1.72 + 0.158i)3-s + (−1.22 + 0.707i)5-s + 0.449·7-s + (2.94 − 0.548i)9-s + 3.14i·11-s + (−3 − 1.73i)13-s + (1.99 − 1.41i)15-s + (−0.550 + 0.317i)17-s + (−3.17 − 2.98i)19-s + (−0.775 + 0.0714i)21-s + (6.12 + 3.53i)23-s + (−1.50 + 2.59i)25-s + (−4.99 + 1.41i)27-s + (1.22 − 2.12i)29-s − 4.24i·31-s + ⋯ |
L(s) = 1 | + (−0.995 + 0.0917i)3-s + (−0.547 + 0.316i)5-s + 0.169·7-s + (0.983 − 0.182i)9-s + 0.948i·11-s + (−0.832 − 0.480i)13-s + (0.516 − 0.365i)15-s + (−0.133 + 0.0770i)17-s + (−0.728 − 0.685i)19-s + (−0.169 + 0.0155i)21-s + (1.27 + 0.737i)23-s + (−0.300 + 0.519i)25-s + (−0.962 + 0.272i)27-s + (0.227 − 0.393i)29-s − 0.762i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.689 + 0.724i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0851393 - 0.198678i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0851393 - 0.198678i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.72 - 0.158i)T \) |
| 19 | \( 1 + (3.17 + 2.98i)T \) |
good | 5 | \( 1 + (1.22 - 0.707i)T + (2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 - 0.449T + 7T^{2} \) |
| 11 | \( 1 - 3.14iT - 11T^{2} \) |
| 13 | \( 1 + (3 + 1.73i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.550 - 0.317i)T + (8.5 - 14.7i)T^{2} \) |
| 23 | \( 1 + (-6.12 - 3.53i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-1.22 + 2.12i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 4.24iT - 31T^{2} \) |
| 37 | \( 1 + 7.70iT - 37T^{2} \) |
| 41 | \( 1 + (1.5 + 2.59i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (4.44 + 7.70i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (11.5 + 6.68i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (5.44 - 9.43i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (5.72 + 9.91i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.775 + 1.34i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.17 + 1.25i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3 + 5.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (4.39 + 7.61i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.34 + 4.24i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 17.0iT - 83T^{2} \) |
| 89 | \( 1 + (-3.55 + 6.14i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.84 + 1.64i)T + (48.5 - 84.0i)T^{2} \) |
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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
−9.889068830973434794813820434283, −9.164382285760969829326689586403, −7.77086916135327617185397335525, −7.22283923559314569643941052440, −6.42589831402705867431444599857, −5.20968726745541476963587163567, −4.64705686724673619814543687516, −3.49723176188910547344448924852, −1.96322193879449765741878637298, −0.12092908824677507958414350416,
1.39882601037000971575206727618, 3.11348767342970864004135789721, 4.51686359058294171846129145716, 4.94889830160876964978098272821, 6.22879404047453459279853661990, 6.78435024665851196463523856237, 7.929790159370875970970571086275, 8.582486166329520361450889065676, 9.731454594583957135198793623823, 10.52207902985180811481071568574