L(s) = 1 | + (−0.258 − 0.965i)2-s + (−1.93 + 0.517i)3-s + (−0.866 + 0.499i)4-s + (0.999 + 1.73i)6-s + (−1.22 − 1.22i)7-s + (0.707 + 0.707i)8-s + (0.866 − 0.499i)9-s + 3·11-s + (1.41 − 1.41i)12-s + (−0.517 + 1.93i)13-s + (−0.866 + 1.49i)14-s + (0.500 − 0.866i)16-s + (−0.707 − 0.707i)18-s + (2.59 + 3.5i)19-s + (2.99 + 1.73i)21-s + (−0.776 − 2.89i)22-s + ⋯ |
L(s) = 1 | + (−0.183 − 0.683i)2-s + (−1.11 + 0.298i)3-s + (−0.433 + 0.249i)4-s + (0.408 + 0.707i)6-s + (−0.462 − 0.462i)7-s + (0.249 + 0.249i)8-s + (0.288 − 0.166i)9-s + 0.904·11-s + (0.408 − 0.408i)12-s + (−0.143 + 0.535i)13-s + (−0.231 + 0.400i)14-s + (0.125 − 0.216i)16-s + (−0.166 − 0.166i)18-s + (0.596 + 0.802i)19-s + (0.654 + 0.377i)21-s + (−0.165 − 0.617i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.566 + 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.251078 - 0.477442i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.251078 - 0.477442i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.258 + 0.965i)T \) |
| 5 | \( 1 \) |
| 19 | \( 1 + (-2.59 - 3.5i)T \) |
good | 3 | \( 1 + (1.93 - 0.517i)T + (2.59 - 1.5i)T^{2} \) |
| 7 | \( 1 + (1.22 + 1.22i)T + 7iT^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (0.517 - 1.93i)T + (-11.2 - 6.5i)T^{2} \) |
| 17 | \( 1 + (14.7 - 8.5i)T^{2} \) |
| 23 | \( 1 + (5.01 + 1.34i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 3.46iT - 31T^{2} \) |
| 37 | \( 1 + (-0.707 + 0.707i)T - 37iT^{2} \) |
| 41 | \( 1 + (4.5 + 2.59i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (1.79 + 6.69i)T + (-37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-2.68 + 10.0i)T + (-40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (0.776 - 2.89i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (-5.19 + 9i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.93 + 0.517i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + (9 + 5.19i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.896 + 3.34i)T + (-63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-1.73 + 3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 83iT^{2} \) |
| 89 | \( 1 + (-2.59 - 4.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.07 - 7.72i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−10.05833785169710241358396434179, −9.210985559949757195928009947081, −8.217873161808356872044736223989, −7.06357244449403390724493284919, −6.23232145452876893455639826795, −5.34523390134475978702146790412, −4.26235062956130890248116802382, −3.54069216300378697859445836982, −1.89881302387080737577765373873, −0.36610345342660147689362679311,
1.14503804391482223726134189553, 3.03724151917399533094724814541, 4.43980468799998130953209957656, 5.44786511862809973662864899462, 6.09465789954059084494611074074, 6.73409783094044537979417146234, 7.60359437503222920707464121234, 8.678393200212373830111268053904, 9.437284472740560154512520371698, 10.23776778737929690982867571832