L(s) = 1 | + (−0.707 + 0.707i)2-s + (−1.32 + 1.32i)3-s − 1.00i·4-s − 1.88i·6-s + (2.26 + 1.36i)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 + (0.855 + 0.517i)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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.916 - 0.400i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.139892 + 0.668988i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.139892 + 0.668988i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.26 - 1.36i)T \) |
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
−11.82139326625894274974684678491, −10.69614862744031425379634430993, −10.20666220788200729834195239895, −9.045799860929417063947396302438, −8.310494075599894081856381534725, −7.08179558111424189071806329367, −6.02673193160185403896204728355, −5.03282651265712400658078695054, −4.26402407768546007035427734271, −1.97274169286209947329944373598,
0.61356851644204905691374325058, 2.04879976697014750568384994600, 3.83288769597674865630917622435, 5.21152361523984502406066679521, 6.33796942097936806409516355025, 7.58086717651188789717636168129, 7.928550826293346103487461987008, 9.474960121083569618118152761160, 10.21234728278846955946204464305, 11.39109490705261452583352868394