L(s) = 1 | + (−0.508 − 1.31i)2-s + (−1.48 + 1.34i)4-s − 2.29·5-s + i·7-s + (2.52 + 1.27i)8-s + (1.16 + 3.03i)10-s − 4.02i·11-s + 1.82i·13-s + (1.31 − 0.508i)14-s + (0.393 − 3.98i)16-s − 0.430i·17-s + 5.01·19-s + (3.40 − 3.08i)20-s + (−5.30 + 2.04i)22-s + 3.49·23-s + ⋯ |
L(s) = 1 | + (−0.359 − 0.933i)2-s + (−0.741 + 0.671i)4-s − 1.02·5-s + 0.377i·7-s + (0.893 + 0.449i)8-s + (0.369 + 0.958i)10-s − 1.21i·11-s + 0.506i·13-s + (0.352 − 0.136i)14-s + (0.0982 − 0.995i)16-s − 0.104i·17-s + 1.14·19-s + (0.761 − 0.689i)20-s + (−1.13 + 0.436i)22-s + 0.728·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.893 - 0.449i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2510584121\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2510584121\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.508 + 1.31i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
good | 5 | \( 1 + 2.29T + 5T^{2} \) |
| 11 | \( 1 + 4.02iT - 11T^{2} \) |
| 13 | \( 1 - 1.82iT - 13T^{2} \) |
| 17 | \( 1 + 0.430iT - 17T^{2} \) |
| 19 | \( 1 - 5.01T + 19T^{2} \) |
| 23 | \( 1 - 3.49T + 23T^{2} \) |
| 29 | \( 1 + 2.16T + 29T^{2} \) |
| 31 | \( 1 - 2.10iT - 31T^{2} \) |
| 37 | \( 1 - 2.19iT - 37T^{2} \) |
| 41 | \( 1 + 4.35iT - 41T^{2} \) |
| 43 | \( 1 + 12.0T + 43T^{2} \) |
| 47 | \( 1 + 1.72T + 47T^{2} \) |
| 53 | \( 1 + 10.0T + 53T^{2} \) |
| 59 | \( 1 + 10.6iT - 59T^{2} \) |
| 61 | \( 1 + 8.44iT - 61T^{2} \) |
| 67 | \( 1 + 12.6T + 67T^{2} \) |
| 71 | \( 1 + 8.95T + 71T^{2} \) |
| 73 | \( 1 + 7.16T + 73T^{2} \) |
| 79 | \( 1 - 15.2iT - 79T^{2} \) |
| 83 | \( 1 + 13.4iT - 83T^{2} \) |
| 89 | \( 1 - 7.44iT - 89T^{2} \) |
| 97 | \( 1 - 1.68T + 97T^{2} \) |
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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
−9.043562073300518489140085176486, −8.338530067449021611656883514778, −7.73105491814355762544991248891, −6.77393717791363088236196721898, −5.47425387203489270218024287701, −4.61081485765494569456066682117, −3.49612739144872000598478121185, −3.05935805849004495537618690597, −1.51947473014610619561532846863, −0.12481655924085978392447974915,
1.38847420706924228112079064989, 3.23564704274264051431244939571, 4.29656432712290827494367015372, 4.93082990814356755375432171757, 5.94407429629899546266379412670, 7.07824590109256688351502630917, 7.45904648559430281624166073569, 8.080575426231219366910600576836, 9.030240786503032320569277552844, 9.817703134717784153331640158517