L(s) = 1 | + (−0.382 + 0.923i)3-s + (3.09 − 1.28i)5-s + (1.73 − 1.73i)7-s + (−0.707 − 0.707i)9-s + (2.39 + 5.79i)11-s + (−0.0173 − 0.00717i)13-s + 3.34i·15-s − 5.57i·17-s + (−1.03 − 0.426i)19-s + (0.938 + 2.26i)21-s + (−2.01 − 2.01i)23-s + (4.39 − 4.39i)25-s + (0.923 − 0.382i)27-s + (0.706 − 1.70i)29-s − 1.38·31-s + ⋯ |
L(s) = 1 | + (−0.220 + 0.533i)3-s + (1.38 − 0.572i)5-s + (0.655 − 0.655i)7-s + (−0.235 − 0.235i)9-s + (0.723 + 1.74i)11-s + (−0.00480 − 0.00199i)13-s + 0.864i·15-s − 1.35i·17-s + (−0.236 − 0.0979i)19-s + (0.204 + 0.494i)21-s + (−0.420 − 0.420i)23-s + (0.878 − 0.878i)25-s + (0.177 − 0.0736i)27-s + (0.131 − 0.316i)29-s − 0.247·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0605i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.98132 + 0.0600428i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.98132 + 0.0600428i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.382 - 0.923i)T \) |
good | 5 | \( 1 + (-3.09 + 1.28i)T + (3.53 - 3.53i)T^{2} \) |
| 7 | \( 1 + (-1.73 + 1.73i)T - 7iT^{2} \) |
| 11 | \( 1 + (-2.39 - 5.79i)T + (-7.77 + 7.77i)T^{2} \) |
| 13 | \( 1 + (0.0173 + 0.00717i)T + (9.19 + 9.19i)T^{2} \) |
| 17 | \( 1 + 5.57iT - 17T^{2} \) |
| 19 | \( 1 + (1.03 + 0.426i)T + (13.4 + 13.4i)T^{2} \) |
| 23 | \( 1 + (2.01 + 2.01i)T + 23iT^{2} \) |
| 29 | \( 1 + (-0.706 + 1.70i)T + (-20.5 - 20.5i)T^{2} \) |
| 31 | \( 1 + 1.38T + 31T^{2} \) |
| 37 | \( 1 + (-2.87 + 1.19i)T + (26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-6.97 - 6.97i)T + 41iT^{2} \) |
| 43 | \( 1 + (-1.67 - 4.03i)T + (-30.4 + 30.4i)T^{2} \) |
| 47 | \( 1 + 1.15iT - 47T^{2} \) |
| 53 | \( 1 + (2.56 + 6.19i)T + (-37.4 + 37.4i)T^{2} \) |
| 59 | \( 1 + (0.735 - 0.304i)T + (41.7 - 41.7i)T^{2} \) |
| 61 | \( 1 + (4.82 - 11.6i)T + (-43.1 - 43.1i)T^{2} \) |
| 67 | \( 1 + (2.05 - 4.97i)T + (-47.3 - 47.3i)T^{2} \) |
| 71 | \( 1 + (-1.78 + 1.78i)T - 71iT^{2} \) |
| 73 | \( 1 + (-1.67 - 1.67i)T + 73iT^{2} \) |
| 79 | \( 1 - 2.67iT - 79T^{2} \) |
| 83 | \( 1 + (6.91 + 2.86i)T + (58.6 + 58.6i)T^{2} \) |
| 89 | \( 1 + (-6.73 + 6.73i)T - 89iT^{2} \) |
| 97 | \( 1 + 1.75T + 97T^{2} \) |
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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
−10.02702011164455166550659543296, −9.651031043165151201273594141802, −8.975872122115281993592816651170, −7.68350614564101260489548172250, −6.78265431366925784449143713134, −5.78935179133040840466844529319, −4.71815743235907823264178091407, −4.36975194045611029067879980485, −2.45837995264677848212891349381, −1.32063561411183292885272599310,
1.43276689342470872466722745162, 2.39332491657838923749411771856, 3.67598901687767383208415143899, 5.35341761881035736116364196440, 6.05335881914798952156094967778, 6.40645415063583079099682432952, 7.81000765352680231478015892587, 8.671377921407283916409819341106, 9.328398137446640205096701926789, 10.57115692494948402203158243140