L(s) = 1 | + (0.707 − 0.707i)2-s − 1.00i·4-s + (−2.23 + 0.0743i)5-s + (−0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s + (−1.52 + 1.63i)10-s − 5.26i·11-s + (−3.16 + 3.16i)13-s − 1.00·14-s − 1.00·16-s + (−3.05 + 3.05i)17-s + (0.0743 + 2.23i)20-s + (−3.72 − 3.72i)22-s + (−4.32 − 4.32i)23-s + (4.98 − 0.332i)25-s + 4.46i·26-s + ⋯ |
L(s) = 1 | + (0.499 − 0.499i)2-s − 0.500i·4-s + (−0.999 + 0.0332i)5-s + (−0.267 − 0.267i)7-s + (−0.250 − 0.250i)8-s + (−0.483 + 0.516i)10-s − 1.58i·11-s + (−0.876 + 0.876i)13-s − 0.267·14-s − 0.250·16-s + (−0.741 + 0.741i)17-s + (0.0166 + 0.499i)20-s + (−0.793 − 0.793i)22-s + (−0.901 − 0.901i)23-s + (0.997 − 0.0664i)25-s + 0.876i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0286i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 630 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0286i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00932317 + 0.650323i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00932317 + 0.650323i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.707 + 0.707i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (2.23 - 0.0743i)T \) |
| 7 | \( 1 + (0.707 + 0.707i)T \) |
good | 11 | \( 1 + 5.26iT - 11T^{2} \) |
| 13 | \( 1 + (3.16 - 3.16i)T - 13iT^{2} \) |
| 17 | \( 1 + (3.05 - 3.05i)T - 17iT^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + (4.32 + 4.32i)T + 23iT^{2} \) |
| 29 | \( 1 + 9.96T + 29T^{2} \) |
| 31 | \( 1 - 1.26T + 31T^{2} \) |
| 37 | \( 1 + (2.93 + 2.93i)T + 37iT^{2} \) |
| 41 | \( 1 + 10.6iT - 41T^{2} \) |
| 43 | \( 1 + (-0.597 + 0.597i)T - 43iT^{2} \) |
| 47 | \( 1 + (-3.42 + 3.42i)T - 47iT^{2} \) |
| 53 | \( 1 + (-9.88 - 9.88i)T + 53iT^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 - 3.05T + 61T^{2} \) |
| 67 | \( 1 + (-4.59 - 4.59i)T + 67iT^{2} \) |
| 71 | \( 1 + 9.23iT - 71T^{2} \) |
| 73 | \( 1 + (-10.2 + 10.2i)T - 73iT^{2} \) |
| 79 | \( 1 - 3.57iT - 79T^{2} \) |
| 83 | \( 1 + (-6.21 - 6.21i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.61T + 89T^{2} \) |
| 97 | \( 1 + (4.63 + 4.63i)T + 97iT^{2} \) |
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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
−10.59170698806987523409278499027, −9.239819040713680651066489574741, −8.530800934608122239677557046955, −7.43958748586646332935634875681, −6.50487267103129642179162864324, −5.48056974459901271704418274865, −4.16136773777423730449853580186, −3.66539121848082707423268610552, −2.27939993445369886582241047100, −0.28325936583748507986899964519,
2.37185519362567684464101658644, 3.64596159764911765178853094983, 4.63658640270750765132705094024, 5.40083821476048761838589386788, 6.78317240102139985223547337702, 7.43993656566036364099583574690, 8.073305328333179002241305680450, 9.324482521377812899739316609038, 10.01638686734235262032906138988, 11.32288563692007447323444528319