L(s) = 1 | − 2.23i·3-s − 2.00·9-s − 2.23i·11-s − 4i·13-s + 3·17-s + 2.23i·19-s + 8.94·23-s − 2.23i·27-s − 4i·29-s + 8.94·31-s − 5.00·33-s + 8i·37-s − 8.94·39-s − 5·41-s − 8.94i·43-s + ⋯ |
L(s) = 1 | − 1.29i·3-s − 0.666·9-s − 0.674i·11-s − 1.10i·13-s + 0.727·17-s + 0.512i·19-s + 1.86·23-s − 0.430i·27-s − 0.742i·29-s + 1.60·31-s − 0.870·33-s + 1.31i·37-s − 1.43·39-s − 0.780·41-s − 1.36i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.887696175\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.887696175\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
good | 3 | \( 1 + 2.23iT - 3T^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + 2.23iT - 11T^{2} \) |
| 13 | \( 1 + 4iT - 13T^{2} \) |
| 17 | \( 1 - 3T + 17T^{2} \) |
| 19 | \( 1 - 2.23iT - 19T^{2} \) |
| 23 | \( 1 - 8.94T + 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 - 8.94T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 5T + 41T^{2} \) |
| 43 | \( 1 + 8.94iT - 43T^{2} \) |
| 47 | \( 1 - 8.94T + 47T^{2} \) |
| 53 | \( 1 - 4iT - 53T^{2} \) |
| 59 | \( 1 - 8.94iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 + 6.70iT - 67T^{2} \) |
| 71 | \( 1 + 8.94T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 + 79T^{2} \) |
| 83 | \( 1 + 6.70iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 - 2T + 97T^{2} \) |
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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
−8.241085043155546754146743018127, −7.64267780013075829648274003253, −6.96747123870673604763334160411, −6.17531760093816033715539489449, −5.57926499046805634258681452068, −4.62288810544811788802928818007, −3.28912358528525055984153250031, −2.72442486495080174502595560066, −1.40466420492133063284002524212, −0.66403475056695630136571874300,
1.31011268342524137903603038099, 2.68425963713436699873147254644, 3.51115545853481729932681872616, 4.52271452759291246942313345114, 4.78286061471826263678987152399, 5.72834279761697451763529212966, 6.83773364390326744705495914983, 7.29004937044603921453560402002, 8.495308028949401692604129683660, 9.083189954188391603352110664339