L(s) = 1 | + (0.783 − 1.17i)2-s + (−0.771 − 1.84i)4-s + (−2.15 − 1.24i)5-s + (−2.64 + 0.0803i)7-s + (−2.77 − 0.537i)8-s + (−3.15 + 1.56i)10-s + (−2.30 − 3.99i)11-s + 5.22·13-s + (−1.97 + 3.17i)14-s + (−2.80 + 2.84i)16-s + (4.85 − 2.80i)17-s + (2.76 + 1.59i)19-s + (−0.632 + 4.93i)20-s + (−6.50 − 0.414i)22-s + (−0.359 + 0.622i)23-s + ⋯ |
L(s) = 1 | + (0.554 − 0.832i)2-s + (−0.385 − 0.922i)4-s + (−0.963 − 0.556i)5-s + (−0.999 + 0.0303i)7-s + (−0.981 − 0.189i)8-s + (−0.997 + 0.493i)10-s + (−0.694 − 1.20i)11-s + 1.44·13-s + (−0.528 + 0.848i)14-s + (−0.702 + 0.712i)16-s + (1.17 − 0.680i)17-s + (0.634 + 0.366i)19-s + (−0.141 + 1.10i)20-s + (−1.38 − 0.0884i)22-s + (−0.0749 + 0.129i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 252 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.921 + 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.209688 - 1.03957i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.209688 - 1.03957i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.783 + 1.17i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.64 - 0.0803i)T \) |
good | 5 | \( 1 + (2.15 + 1.24i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.30 + 3.99i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.22T + 13T^{2} \) |
| 17 | \( 1 + (-4.85 + 2.80i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.76 - 1.59i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.359 - 0.622i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 4.53iT - 29T^{2} \) |
| 31 | \( 1 + (1.01 - 0.588i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.35 - 2.34i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 3.83iT - 41T^{2} \) |
| 43 | \( 1 + 11.1iT - 43T^{2} \) |
| 47 | \( 1 + (2.70 - 4.69i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (1.79 - 1.03i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.05 - 3.56i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-0.505 + 0.874i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-10.9 + 6.32i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 7.31T + 71T^{2} \) |
| 73 | \( 1 + (4.81 + 8.34i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.65 - 4.41i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 13.7T + 83T^{2} \) |
| 89 | \( 1 + (-7.38 - 4.26i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 10.7T + 97T^{2} \) |
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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
−11.75503892799281934024189327956, −10.88964219726641905792256152615, −9.893480860231355554367795316734, −8.834322715416194860440097978412, −7.88795150620904203687916854406, −6.21143064227001165915447680692, −5.33629170765633469672922870886, −3.79737444427779901077985940594, −3.16022936584994484710479274289, −0.73163621906396291923538185115,
3.17266710041167715130673589629, 3.93472447448511207087617642944, 5.41193432127213858179047149975, 6.56736149943911477492347228768, 7.39538326847584953126832230389, 8.207087410342692906475063971219, 9.468097306390400674025839392731, 10.60642807810275670981954619514, 11.76194224933894709704438019775, 12.66907779282013250458255348970