| L(s) = 1 | + (3 − 3i)3-s + (−64.8 − 64.8i)7-s + 63i·9-s + 33.7·11-s + (−132. + 132. i)13-s + (270. + 270. i)17-s + 295. i·19-s − 389.·21-s + (−1.44 + 1.44i)23-s + (432 + 432i)27-s + 297. i·29-s − 1.22e3·31-s + (101. − 101. i)33-s + (−402. − 402. i)37-s + 797. i·39-s + ⋯ |
| L(s) = 1 | + (0.333 − 0.333i)3-s + (−1.32 − 1.32i)7-s + 0.777i·9-s + 0.278·11-s + (−0.786 + 0.786i)13-s + (0.935 + 0.935i)17-s + 0.819i·19-s − 0.882·21-s + (−0.00273 + 0.00273i)23-s + (0.592 + 0.592i)27-s + 0.353i·29-s − 1.27·31-s + (0.0928 − 0.0928i)33-s + (−0.293 − 0.293i)37-s + 0.524i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 200 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.8391711304\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8391711304\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-3 + 3i)T - 81iT^{2} \) |
| 7 | \( 1 + (64.8 + 64.8i)T + 2.40e3iT^{2} \) |
| 11 | \( 1 - 33.7T + 1.46e4T^{2} \) |
| 13 | \( 1 + (132. - 132. i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (-270. - 270. i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 - 295. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (1.44 - 1.44i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 - 297. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.22e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + (402. + 402. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 - 860.T + 2.82e6T^{2} \) |
| 43 | \( 1 + (753. - 753. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-2.22e3 - 2.22e3i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (2.53e3 - 2.53e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + 3.42e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 6.69e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-318. - 318. i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 + 3.42e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (3.05e3 - 3.05e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 - 2.63e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (225. - 225. i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 + 4.98e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (4.89e3 + 4.89e3i)T + 8.85e7iT^{2} \) |
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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
−12.41081273464281095165987009228, −10.87335582079514409019861022036, −10.12602489292295835973269073975, −9.221513951678434287420271754064, −7.76027080348157763357212348764, −7.13931694838132321666124239493, −5.98144964697240152300765097885, −4.31388281171623696253127698246, −3.22289677007674935477358898514, −1.53516113652815665322984057231,
0.27973126508083906074520844002, 2.65640934851210244953993566660, 3.44735659788003820365513379003, 5.20919109537640567259281003386, 6.21169657175480113294324945555, 7.36112016982547799977491814140, 8.902098910257098687576650753092, 9.394276384009051729124803040364, 10.21254285992670388123008799763, 11.84167740652236874579533003091
