L(s) = 1 | + (1.23 − 1.86i)5-s + (3.46 + 2i)7-s + (2 − 3.46i)11-s − 4i·17-s + (3.46 − 2i)23-s + (−1.96 − 4.59i)25-s + (−3 + 5.19i)29-s + (−2 − 3.46i)31-s + (8 − 4i)35-s + 8i·37-s + (5 + 8.66i)41-s + (−3.46 − 2i)43-s + (−3.46 − 2i)47-s + (4.49 + 7.79i)49-s − 12i·53-s + ⋯ |
L(s) = 1 | + (0.550 − 0.834i)5-s + (1.30 + 0.755i)7-s + (0.603 − 1.04i)11-s − 0.970i·17-s + (0.722 − 0.417i)23-s + (−0.392 − 0.919i)25-s + (−0.557 + 0.964i)29-s + (−0.359 − 0.622i)31-s + (1.35 − 0.676i)35-s + 1.31i·37-s + (0.780 + 1.35i)41-s + (−0.528 − 0.304i)43-s + (−0.505 − 0.291i)47-s + (0.642 + 1.11i)49-s − 1.64i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.285098398\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.285098398\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-1.23 + 1.86i)T \) |
good | 7 | \( 1 + (-3.46 - 2i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2 + 3.46i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + (-3.46 + 2i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + (-5 - 8.66i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.46 + 2i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (3.46 + 2i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 + 2i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 8iT - 73T^{2} \) |
| 79 | \( 1 + (6 - 10.3i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.46 + 2i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 + (-6.92 - 4i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.124410620979882437952675262575, −8.567472859117818704247847468861, −8.009004515271222247859368282561, −6.82093298763815360226846357120, −5.85522153144080576611243305418, −5.15051545039247505311528480346, −4.57599004863505965373510384617, −3.18987741052472924101280296814, −1.99758018313292688070117116719, −1.00022395715148385726328433653,
1.44872113748713862608872106888, 2.20587067679222812709681630030, 3.66847953874837885860324728168, 4.41055088770428777547438274640, 5.40338262845794400862969989394, 6.31445654900072427012460982550, 7.32278411839072695737823594711, 7.56916942861005585525355350340, 8.753917271321828150628947815120, 9.556100248431572131727442606771