L(s) = 1 | + (−1.37 − 0.340i)2-s + (1.76 + 0.934i)4-s + (2.75 − 2.75i)5-s + 0.113i·7-s + (−2.10 − 1.88i)8-s + (−4.71 + 2.84i)10-s + (3.69 − 3.69i)11-s + (−1.53 − 1.53i)13-s + (0.0386 − 0.155i)14-s + (2.25 + 3.30i)16-s − 6.62·17-s + (3.27 + 3.27i)19-s + (7.43 − 2.29i)20-s + (−6.32 + 3.81i)22-s + 6.20i·23-s + ⋯ |
L(s) = 1 | + (−0.970 − 0.240i)2-s + (0.884 + 0.467i)4-s + (1.23 − 1.23i)5-s + 0.0428i·7-s + (−0.745 − 0.666i)8-s + (−1.49 + 0.898i)10-s + (1.11 − 1.11i)11-s + (−0.426 − 0.426i)13-s + (0.0103 − 0.0416i)14-s + (0.563 + 0.826i)16-s − 1.60·17-s + (0.751 + 0.751i)19-s + (1.66 − 0.512i)20-s + (−1.34 + 0.812i)22-s + 1.29i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.203 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.203 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.858096 - 0.697726i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.858096 - 0.697726i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1.37 + 0.340i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (-2.75 + 2.75i)T - 5iT^{2} \) |
| 7 | \( 1 - 0.113iT - 7T^{2} \) |
| 11 | \( 1 + (-3.69 + 3.69i)T - 11iT^{2} \) |
| 13 | \( 1 + (1.53 + 1.53i)T + 13iT^{2} \) |
| 17 | \( 1 + 6.62T + 17T^{2} \) |
| 19 | \( 1 + (-3.27 - 3.27i)T + 19iT^{2} \) |
| 23 | \( 1 - 6.20iT - 23T^{2} \) |
| 29 | \( 1 + (1.51 + 1.51i)T + 29iT^{2} \) |
| 31 | \( 1 + 1.10T + 31T^{2} \) |
| 37 | \( 1 + (-2.61 + 2.61i)T - 37iT^{2} \) |
| 41 | \( 1 + 9.18iT - 41T^{2} \) |
| 43 | \( 1 + (-3.18 + 3.18i)T - 43iT^{2} \) |
| 47 | \( 1 - 3.56T + 47T^{2} \) |
| 53 | \( 1 + (4.08 - 4.08i)T - 53iT^{2} \) |
| 59 | \( 1 + (0.942 - 0.942i)T - 59iT^{2} \) |
| 61 | \( 1 + (0.437 + 0.437i)T + 61iT^{2} \) |
| 67 | \( 1 + (3.26 + 3.26i)T + 67iT^{2} \) |
| 71 | \( 1 - 7.72iT - 71T^{2} \) |
| 73 | \( 1 - 10.2iT - 73T^{2} \) |
| 79 | \( 1 + 1.25T + 79T^{2} \) |
| 83 | \( 1 + (-8.42 - 8.42i)T + 83iT^{2} \) |
| 89 | \( 1 - 0.361iT - 89T^{2} \) |
| 97 | \( 1 - 13.7T + 97T^{2} \) |
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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
−10.83741088909563686000702291826, −9.775630931629071442685013902160, −9.091879998183637264884798421099, −8.694570397536089994099412097622, −7.44430544445991097890768360293, −6.17560770210044260551592546030, −5.50078717784014812717346507211, −3.85382861536305477739461529643, −2.19772669960918685174310980164, −1.01730008165747032268043962990,
1.84492168534468760082327728717, 2.71719715972749628879088700045, 4.69412009688897322688233577480, 6.27328371941254913429192786670, 6.68644308186683613191994852763, 7.41547461369417164796281304666, 9.042271962772386921445126882050, 9.447408644146220030661548130529, 10.30254117755088387325839497065, 11.05685827119729650738595485261