| L(s) = 1 | + 3i·3-s − 6i·5-s − 16·7-s − 9·9-s + 12i·11-s + 38i·13-s + 18·15-s − 126·17-s − 20i·19-s − 48i·21-s + 168·23-s + 89·25-s − 27i·27-s + 30i·29-s + 88·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.536i·5-s − 0.863·7-s − 0.333·9-s + 0.328i·11-s + 0.810i·13-s + 0.309·15-s − 1.79·17-s − 0.241i·19-s − 0.498i·21-s + 1.52·23-s + 0.711·25-s − 0.192i·27-s + 0.192i·29-s + 0.509·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.230811391\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.230811391\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3iT \) |
| good | 5 | \( 1 + 6iT - 125T^{2} \) |
| 7 | \( 1 + 16T + 343T^{2} \) |
| 11 | \( 1 - 12iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 38iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 126T + 4.91e3T^{2} \) |
| 19 | \( 1 + 20iT - 6.85e3T^{2} \) |
| 23 | \( 1 - 168T + 1.21e4T^{2} \) |
| 29 | \( 1 - 30iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 88T + 2.97e4T^{2} \) |
| 37 | \( 1 + 254iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 42T + 6.89e4T^{2} \) |
| 43 | \( 1 + 52iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 96T + 1.03e5T^{2} \) |
| 53 | \( 1 + 198iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 660iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 538iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 884iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 792T + 3.57e5T^{2} \) |
| 73 | \( 1 + 218T + 3.89e5T^{2} \) |
| 79 | \( 1 - 520T + 4.93e5T^{2} \) |
| 83 | \( 1 - 492iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 810T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.15e3T + 9.12e5T^{2} \) |
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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
−9.498111105333350892084779592978, −9.214618776931419988546075362463, −8.398935249589196886577598331979, −6.95812402953738979044416973864, −6.47715174776521429248282211237, −5.08333347813586004880108509639, −4.46602022798765925081095274529, −3.35154868152655505735652216512, −2.13195738594833807605148774066, −0.41852356046069948294231679342,
0.905440116914145266160422663682, 2.56142429047478835611354208653, 3.21528388210715666629842769754, 4.61504904368624792666605264565, 5.84910501489744073833959044720, 6.66585938539681730376616789586, 7.18970957262973087806883537088, 8.398805403693490735520427270585, 9.052093973087600044773336348302, 10.14954254829679674780230747533
