L(s) = 1 | + (0.707 − 0.707i)2-s + (−1.32 + 1.32i)3-s − 1.00i·4-s + 1.88i·6-s + (1.36 + 2.26i)7-s + (−0.707 − 0.707i)8-s − 0.535i·9-s + 1.73·11-s + (1.32 + 1.32i)12-s + (−3.63 + 3.63i)13-s + (2.56 + 0.633i)14-s − 1.00·16-s + (2.30 + 2.30i)17-s + (−0.378 − 0.378i)18-s + 3.25·19-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s + (−0.767 + 0.767i)3-s − 0.500i·4-s + 0.767i·6-s + (0.517 + 0.855i)7-s + (−0.250 − 0.250i)8-s − 0.178i·9-s + 0.522·11-s + (0.383 + 0.383i)12-s + (−1.00 + 1.00i)13-s + (0.686 + 0.169i)14-s − 0.250·16-s + (0.558 + 0.558i)17-s + (−0.0893 − 0.0893i)18-s + 0.747·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.640 - 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.22027 + 0.571448i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.22027 + 0.571448i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.36 - 2.26i)T \) |
good | 3 | \( 1 + (1.32 - 1.32i)T - 3iT^{2} \) |
| 11 | \( 1 - 1.73T + 11T^{2} \) |
| 13 | \( 1 + (3.63 - 3.63i)T - 13iT^{2} \) |
| 17 | \( 1 + (-2.30 - 2.30i)T + 17iT^{2} \) |
| 19 | \( 1 - 3.25T + 19T^{2} \) |
| 23 | \( 1 + (-5.79 - 5.79i)T + 23iT^{2} \) |
| 29 | \( 1 - 4.73iT - 29T^{2} \) |
| 31 | \( 1 + 8.89iT - 31T^{2} \) |
| 37 | \( 1 + (-1.55 + 1.55i)T - 37iT^{2} \) |
| 41 | \( 1 - 5.64iT - 41T^{2} \) |
| 43 | \( 1 + (2.44 + 2.44i)T + 43iT^{2} \) |
| 47 | \( 1 + (6.29 + 6.29i)T + 47iT^{2} \) |
| 53 | \( 1 + (10.0 + 10.0i)T + 53iT^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 6.51iT - 61T^{2} \) |
| 67 | \( 1 + (-7.02 + 7.02i)T - 67iT^{2} \) |
| 71 | \( 1 - 8.19T + 71T^{2} \) |
| 73 | \( 1 + (-7.62 + 7.62i)T - 73iT^{2} \) |
| 79 | \( 1 - 2iT - 79T^{2} \) |
| 83 | \( 1 + (-3.98 + 3.98i)T - 83iT^{2} \) |
| 89 | \( 1 + 5.64T + 89T^{2} \) |
| 97 | \( 1 + (-7.26 - 7.26i)T + 97iT^{2} \) |
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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
−11.51215708010502603439019002600, −11.07343388473527943239338116266, −9.735007692914236823307477998848, −9.358437979042100946432487935472, −7.85464531111582443357213437794, −6.48416329169891465763998205976, −5.29749715604075918803064947048, −4.86312964463564479848111931220, −3.54396450783336187426042994721, −1.90562284567538727278292147327,
0.951048946975474971975529853595, 3.10145295318276438986256442731, 4.65484154818295144124209241475, 5.46745990138002377353468420054, 6.68594276460573538431174773141, 7.26812732031327559785906219580, 8.124859934780388011629344663751, 9.542447946792740919180206690429, 10.68213633943918633126720812683, 11.59448378494578098140460388792