L(s) = 1 | + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (7 + 12.1i)5-s + 7.99·8-s + (14 − 24.2i)10-s + (−14 + 24.2i)11-s − 18·13-s + (−8 − 13.8i)16-s + (−37 + 64.0i)17-s + (40 + 69.2i)19-s − 56·20-s + 56·22-s + (−56 − 96.9i)23-s + (−35.5 + 61.4i)25-s + (18 + 31.1i)26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.626 + 1.08i)5-s + 0.353·8-s + (0.442 − 0.766i)10-s + (−0.383 + 0.664i)11-s − 0.384·13-s + (−0.125 − 0.216i)16-s + (−0.527 + 0.914i)17-s + (0.482 + 0.836i)19-s − 0.626·20-s + 0.542·22-s + (−0.507 − 0.879i)23-s + (−0.284 + 0.491i)25-s + (0.135 + 0.235i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.4632870513\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4632870513\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-7 - 12.1i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (14 - 24.2i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + 18T + 2.19e3T^{2} \) |
| 17 | \( 1 + (37 - 64.0i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-40 - 69.2i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (56 + 96.9i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 + 190T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-36 + 62.3i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-173 - 299. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 162T + 6.89e4T^{2} \) |
| 43 | \( 1 + 412T + 7.95e4T^{2} \) |
| 47 | \( 1 + (12 + 20.7i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-159 + 275. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-100 + 173. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (99 + 171. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-358 + 620. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 + 392T + 3.57e5T^{2} \) |
| 73 | \( 1 + (-269 + 465. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (120 + 207. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.07e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (405 + 701. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 1.35e3T + 9.12e5T^{2} \) |
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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
−10.00170122087802948614579645946, −9.774911121411751586907822049782, −8.477209533358224688985791900048, −7.68455546248623220390012617973, −6.73736918946184096340636719163, −5.93721407569318898016079678301, −4.68055355915688163262178823103, −3.55752672101578901233160434951, −2.49323570940076165265261769654, −1.73621844186879382633949984982,
0.13465366230201023326126197245, 1.25762082007087556917051727696, 2.60850489057279171447232983818, 4.18864977934043127024094913692, 5.31457054639016316857282121875, 5.60100855273159081277578264392, 6.89365759487603518281850144934, 7.68447865745675963982812829219, 8.642471081015980497833945130408, 9.300491288941149491459526589129