| L(s) = 1 | + (−2.08 − 2.08i)2-s − 7.34i·4-s + (−65.1 − 65.1i)7-s + (−48.5 + 48.5i)8-s − 56.3·11-s + (−0.983 + 0.983i)13-s + 270. i·14-s + 84.5·16-s + (159. + 159. i)17-s − 265. i·19-s + (117. + 117. i)22-s + (−185. + 185. i)23-s + 4.09·26-s + (−478. + 478. i)28-s − 544. i·29-s + ⋯ |
| L(s) = 1 | + (−0.520 − 0.520i)2-s − 0.458i·4-s + (−1.32 − 1.32i)7-s + (−0.758 + 0.758i)8-s − 0.466·11-s + (−0.00582 + 0.00582i)13-s + 1.38i·14-s + 0.330·16-s + (0.550 + 0.550i)17-s − 0.735i·19-s + (0.242 + 0.242i)22-s + (−0.350 + 0.350i)23-s + 0.00605·26-s + (−0.609 + 0.609i)28-s − 0.646i·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.1743190692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1743190692\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (2.08 + 2.08i)T + 16iT^{2} \) |
| 7 | \( 1 + (65.1 + 65.1i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 + 56.3T + 1.46e4T^{2} \) |
| 13 | \( 1 + (0.983 - 0.983i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (-159. - 159. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 + 265. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (185. - 185. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 + 544. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 710.T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-639. - 639. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 - 1.32e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (-22.3 + 22.3i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-456. - 456. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (424. - 424. i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 - 3.46e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 4.93e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-3.80e3 - 3.80e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + 7.95e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (-1.94e3 + 1.94e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 4.08e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (9.10e3 - 9.10e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 7.01e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (2.57e3 + 2.57e3i)T + 8.85e7iT^{2} \) |
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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
−11.45222501465408269762188349574, −10.51489295124273210211835031722, −9.975477818898610219476387135452, −9.169601970197417304479441274306, −7.79146068435597227067716914802, −6.68436601123241589007720708443, −5.64339475936144233755128109362, −4.04126945161575300901646907824, −2.74921369368079083039937945779, −1.02694208748196626896720807939,
0.083279896033383397229737299648, 2.57956052442528649948865251058, 3.58557858270766126286829361011, 5.51203242952074910500108435854, 6.38993556941815886621816845688, 7.46384824915154522452721194081, 8.520599209744656078960807002807, 9.316999191114424971501926239302, 10.06406455067532308952548729181, 11.62157421185876572572653612470
