L(s) = 1 | + i·7-s + 4.24i·11-s + 7i·13-s − 4.24·17-s − 7·19-s − 29.6·23-s − 29.6i·29-s − 17·31-s − 16i·37-s − 50.9i·41-s − 55i·43-s + 46.6·47-s + 48·49-s + 84.8·53-s + 55.1i·59-s + ⋯ |
L(s) = 1 | + 0.142i·7-s + 0.385i·11-s + 0.538i·13-s − 0.249·17-s − 0.368·19-s − 1.29·23-s − 1.02i·29-s − 0.548·31-s − 0.432i·37-s − 1.24i·41-s − 1.27i·43-s + 0.992·47-s + 0.979·49-s + 1.60·53-s + 0.934i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.881 + 0.472i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.653537321\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.653537321\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 - iT - 49T^{2} \) |
| 11 | \( 1 - 4.24iT - 121T^{2} \) |
| 13 | \( 1 - 7iT - 169T^{2} \) |
| 17 | \( 1 + 4.24T + 289T^{2} \) |
| 19 | \( 1 + 7T + 361T^{2} \) |
| 23 | \( 1 + 29.6T + 529T^{2} \) |
| 29 | \( 1 + 29.6iT - 841T^{2} \) |
| 31 | \( 1 + 17T + 961T^{2} \) |
| 37 | \( 1 + 16iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 50.9iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 55iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 46.6T + 2.20e3T^{2} \) |
| 53 | \( 1 - 84.8T + 2.80e3T^{2} \) |
| 59 | \( 1 - 55.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 65T + 3.72e3T^{2} \) |
| 67 | \( 1 - 49iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 50.9iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 88iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 40T + 6.24e3T^{2} \) |
| 83 | \( 1 + 156.T + 6.88e3T^{2} \) |
| 89 | \( 1 + 101. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 41iT - 9.40e3T^{2} \) |
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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
−8.605997402989778148866095046958, −7.34118781726713713358044458926, −7.08216726051260344056019456522, −5.91239658913113861251321831736, −5.51582434385837275262712927314, −4.18269426792325219441721130030, −3.98501714793386834719198359304, −2.49834403009239417562583158817, −1.92799397380963088445066090780, −0.48512447897100891555750904501,
0.73229931898736577015297534001, 1.89868580307932402102721322859, 2.91662175476413268462561485517, 3.77383342142740767800209859770, 4.59179299112211569662863766498, 5.50738586089344597633639852899, 6.16636741472943599436112381352, 6.95676102765559718176687214050, 7.78445595176875723064585921832, 8.384144099651074709846519790837