L(s) = 1 | + (30.5 − 30.5i)2-s + (−129. + 54.5i)3-s − 1.35e3i·4-s + (−1.26e3 + 602. i)5-s + (−2.28e3 + 5.60e3i)6-s + (−2.74e3 − 2.74e3i)7-s + (−2.56e4 − 2.56e4i)8-s + (1.37e4 − 1.40e4i)9-s + (−2.01e4 + 5.68e4i)10-s + 4.81e3i·11-s + (7.36e4 + 1.74e5i)12-s + (6.67e4 − 6.67e4i)13-s − 1.67e5·14-s + (1.30e5 − 1.46e5i)15-s − 8.71e5·16-s + (3.46e4 − 3.46e4i)17-s + ⋯ |
L(s) = 1 | + (1.34 − 1.34i)2-s + (−0.921 + 0.388i)3-s − 2.63i·4-s + (−0.902 + 0.430i)5-s + (−0.718 + 1.76i)6-s + (−0.432 − 0.432i)7-s + (−2.21 − 2.21i)8-s + (0.698 − 0.715i)9-s + (−0.635 + 1.79i)10-s + 0.0991i·11-s + (1.02 + 2.43i)12-s + (0.648 − 0.648i)13-s − 1.16·14-s + (0.664 − 0.747i)15-s − 3.32·16-s + (0.100 − 0.100i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.171i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.171i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(5)\) |
\(\approx\) |
\(0.132163 + 1.52688i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.132163 + 1.52688i\) |
\(L(\frac{11}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (129. - 54.5i)T \) |
| 5 | \( 1 + (1.26e3 - 602. i)T \) |
good | 2 | \( 1 + (-30.5 + 30.5i)T - 512iT^{2} \) |
| 7 | \( 1 + (2.74e3 + 2.74e3i)T + 4.03e7iT^{2} \) |
| 11 | \( 1 - 4.81e3iT - 2.35e9T^{2} \) |
| 13 | \( 1 + (-6.67e4 + 6.67e4i)T - 1.06e10iT^{2} \) |
| 17 | \( 1 + (-3.46e4 + 3.46e4i)T - 1.18e11iT^{2} \) |
| 19 | \( 1 + 4.27e5iT - 3.22e11T^{2} \) |
| 23 | \( 1 + (6.70e5 + 6.70e5i)T + 1.80e12iT^{2} \) |
| 29 | \( 1 + 8.83e5T + 1.45e13T^{2} \) |
| 31 | \( 1 - 7.22e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + (5.43e6 + 5.43e6i)T + 1.29e14iT^{2} \) |
| 41 | \( 1 - 2.76e7iT - 3.27e14T^{2} \) |
| 43 | \( 1 + (1.38e7 - 1.38e7i)T - 5.02e14iT^{2} \) |
| 47 | \( 1 + (1.97e6 - 1.97e6i)T - 1.11e15iT^{2} \) |
| 53 | \( 1 + (-1.15e7 - 1.15e7i)T + 3.29e15iT^{2} \) |
| 59 | \( 1 + 2.00e7T + 8.66e15T^{2} \) |
| 61 | \( 1 - 1.28e8T + 1.16e16T^{2} \) |
| 67 | \( 1 + (9.09e7 + 9.09e7i)T + 2.72e16iT^{2} \) |
| 71 | \( 1 + 4.40e7iT - 4.58e16T^{2} \) |
| 73 | \( 1 + (-3.10e8 + 3.10e8i)T - 5.88e16iT^{2} \) |
| 79 | \( 1 + 3.03e8iT - 1.19e17T^{2} \) |
| 83 | \( 1 + (4.46e8 + 4.46e8i)T + 1.86e17iT^{2} \) |
| 89 | \( 1 - 1.06e9T + 3.50e17T^{2} \) |
| 97 | \( 1 + (7.40e8 + 7.40e8i)T + 7.60e17iT^{2} \) |
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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
−15.97340045639437394292169053924, −14.94023815380526191066468351301, −13.27400505055870406278268423500, −12.05606404050965148992132102443, −11.06801914570780072584587317787, −10.10885863353521849253376331135, −6.38856078097919595558850424611, −4.62278214533908473450070609028, −3.34299702250280626020430318322, −0.59288851928716532929711517672,
3.97829937909448120232360699374, 5.51371868466814903933358804095, 6.80316733557740327155789335475, 8.237148201616436733649893462545, 11.70689839497678131094344395692, 12.50114891600751844868904197898, 13.75169048647412546761672495297, 15.50891163955098891147699759640, 16.18598398313784609953725642872, 17.19705648138105548850389285333