L(s) = 1 | + (0.287 − 0.287i)2-s + 7.83i·4-s + (−15.0 − 15.0i)7-s + (4.55 + 4.55i)8-s + 27.9i·11-s + (−2.49 + 2.49i)13-s − 8.66·14-s − 60.0·16-s + (−67.9 + 67.9i)17-s − 95.2i·19-s + (8.06 + 8.06i)22-s + (−121. − 121. i)23-s + 1.43i·26-s + (117. − 117. i)28-s − 99.0·29-s + ⋯ |
L(s) = 1 | + (0.101 − 0.101i)2-s + 0.979i·4-s + (−0.812 − 0.812i)7-s + (0.201 + 0.201i)8-s + 0.767i·11-s + (−0.0533 + 0.0533i)13-s − 0.165·14-s − 0.938·16-s + (−0.968 + 0.968i)17-s − 1.14i·19-s + (0.0781 + 0.0781i)22-s + (−1.10 − 1.10i)23-s + 0.0108i·26-s + (0.796 − 0.796i)28-s − 0.634·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0618i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.00739727 - 0.238910i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00739727 - 0.238910i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + (-0.287 + 0.287i)T - 8iT^{2} \) |
| 7 | \( 1 + (15.0 + 15.0i)T + 343iT^{2} \) |
| 11 | \( 1 - 27.9iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (2.49 - 2.49i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (67.9 - 67.9i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 95.2iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (121. + 121. i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 + 99.0T + 2.43e4T^{2} \) |
| 31 | \( 1 + 28.7T + 2.97e4T^{2} \) |
| 37 | \( 1 + (271. + 271. i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 453. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (30.5 - 30.5i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (254. - 254. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (-224. - 224. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 483.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 264.T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-498. - 498. i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 609. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-74.6 + 74.6i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 - 406. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (-652. - 652. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 + 139.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (557. + 557. i)T + 9.12e5iT^{2} \) |
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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
−12.55618718655849736393754494067, −11.35170706214859826294377257363, −10.42973589451127723733491503230, −9.347750367118054561541409719628, −8.301455801611637970331641648789, −7.19499462767845283454670133818, −6.47277406517356902510339937860, −4.57239808513529315748177459594, −3.73472840039101738608881805274, −2.30633632096006743832357387893,
0.088076601131234593469996308025, 2.00419137400384654120926013798, 3.59482302884292044209127898460, 5.27400936542543728175359058315, 5.98143251388905418618711868763, 6.99071578611981920319664666805, 8.521796915159794971599108613818, 9.454527769517227910491326211789, 10.19687205897481499636946707155, 11.33733515074787545340774133216