L(s) = 1 | + (0.653 − 0.653i)2-s + (−0.707 + 0.707i)3-s + 1.14i·4-s + 0.924i·6-s + (−2.53 + 0.741i)7-s + (2.05 + 2.05i)8-s − 1.00i·9-s − 0.746·11-s + (−0.809 − 0.809i)12-s + (−4.26 + 4.26i)13-s + (−1.17 + 2.14i)14-s + 0.398·16-s + (−5.29 − 5.29i)17-s + (−0.653 − 0.653i)18-s + 1.17·19-s + ⋯ |
L(s) = 1 | + (0.462 − 0.462i)2-s + (−0.408 + 0.408i)3-s + 0.572i·4-s + 0.377i·6-s + (−0.959 + 0.280i)7-s + (0.726 + 0.726i)8-s − 0.333i·9-s − 0.225·11-s + (−0.233 − 0.233i)12-s + (−1.18 + 1.18i)13-s + (−0.314 + 0.573i)14-s + 0.0996·16-s + (−1.28 − 1.28i)17-s + (−0.154 − 0.154i)18-s + 0.269·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 525 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.578 - 0.815i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.404916 + 0.783165i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.404916 + 0.783165i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.707 - 0.707i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (2.53 - 0.741i)T \) |
good | 2 | \( 1 + (-0.653 + 0.653i)T - 2iT^{2} \) |
| 11 | \( 1 + 0.746T + 11T^{2} \) |
| 13 | \( 1 + (4.26 - 4.26i)T - 13iT^{2} \) |
| 17 | \( 1 + (5.29 + 5.29i)T + 17iT^{2} \) |
| 19 | \( 1 - 1.17T + 19T^{2} \) |
| 23 | \( 1 + (-3.19 - 3.19i)T + 23iT^{2} \) |
| 29 | \( 1 - 2.45iT - 29T^{2} \) |
| 31 | \( 1 - 4.87iT - 31T^{2} \) |
| 37 | \( 1 + (2.05 - 2.05i)T - 37iT^{2} \) |
| 41 | \( 1 - 6.92iT - 41T^{2} \) |
| 43 | \( 1 + (6.64 + 6.64i)T + 43iT^{2} \) |
| 47 | \( 1 + (-5.29 - 5.29i)T + 47iT^{2} \) |
| 53 | \( 1 + (-4.89 - 4.89i)T + 53iT^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 + 2.52iT - 61T^{2} \) |
| 67 | \( 1 + (-6.47 + 6.47i)T - 67iT^{2} \) |
| 71 | \( 1 + 6.74T + 71T^{2} \) |
| 73 | \( 1 + (2.47 - 2.47i)T - 73iT^{2} \) |
| 79 | \( 1 + 5.83iT - 79T^{2} \) |
| 83 | \( 1 + (-0.768 + 0.768i)T - 83iT^{2} \) |
| 89 | \( 1 + 3.69T + 89T^{2} \) |
| 97 | \( 1 + (-10.2 - 10.2i)T + 97iT^{2} \) |
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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
−11.49953851628960516264394918726, −10.38844242855379646602621544983, −9.405410969551715877892296443746, −8.822985716955951213353796819930, −7.25819811675907285987953348444, −6.75766622573574892233590636066, −5.20460132308449769982955095989, −4.53958238621606924708201312131, −3.32717555720815655239339419385, −2.37163297985908340522207832578,
0.43974529540580242028740691427, 2.37204134740759158278156248833, 3.98220278952373949011693054264, 5.12075581297403790413260722541, 5.94354448898471137596856231022, 6.76587030520966786317252119807, 7.45708856887673188513554707906, 8.713863346333704501234597364085, 10.05739784608306946634307577705, 10.31838472710491008714834243848