L(s) = 1 | + (4.54 − 2.52i)3-s + 8.01·5-s + 12.6i·7-s + (14.2 − 22.9i)9-s − 37.7i·11-s − 60.0i·13-s + (36.3 − 20.2i)15-s − 37.0i·17-s − 127.·19-s + (31.7 + 57.2i)21-s + 56.9·23-s − 60.8·25-s + (7.09 − 140. i)27-s − 220.·29-s + 2.26i·31-s + ⋯ |
L(s) = 1 | + (0.874 − 0.485i)3-s + 0.716·5-s + 0.680i·7-s + (0.528 − 0.848i)9-s − 1.03i·11-s − 1.28i·13-s + (0.626 − 0.347i)15-s − 0.528i·17-s − 1.54·19-s + (0.330 + 0.594i)21-s + 0.516·23-s − 0.486·25-s + (0.0505 − 0.998i)27-s − 1.41·29-s + 0.0130i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.275 + 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.685624304\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.685624304\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-4.54 + 2.52i)T \) |
good | 5 | \( 1 - 8.01T + 125T^{2} \) |
| 7 | \( 1 - 12.6iT - 343T^{2} \) |
| 11 | \( 1 + 37.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 60.0iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 37.0iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 127.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 56.9T + 1.21e4T^{2} \) |
| 29 | \( 1 + 220.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 2.26iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 166. iT - 5.06e4T^{2} \) |
| 41 | \( 1 - 154. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 53.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 591.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 538.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 586. iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 431. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 175.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 29.5T + 3.57e5T^{2} \) |
| 73 | \( 1 + 937.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 409. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.29e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 921. iT - 7.04e5T^{2} \) |
| 97 | \( 1 + 1.64e3T + 9.12e5T^{2} \) |
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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
−9.451460360414111385651130244696, −8.795678724950963026067952184714, −8.129725608132822806802240095627, −7.16203065416470900460206540369, −6.03210539461211531688657686885, −5.50565797434989460070294024388, −3.90230327579893887687625657722, −2.82361318553427959456775786463, −2.05722794200571935986277694954, −0.59511020074105163608701731188,
1.68493661170337251585813034052, 2.36867958143992135725280094741, 4.01519633666489850749440393820, 4.35749156884370187123589093223, 5.72018015670892799155419968336, 6.91691601768184926782028854765, 7.51292455366083680318357729119, 8.795692930955702857554204747657, 9.215410221100734721541348533172, 10.24174167347906314043732281918