L(s) = 1 | + 2.14·3-s − 1.60·5-s − 2.89·7-s + 1.60·9-s + 11-s + 4.89·13-s − 3.43·15-s − 5.74·17-s + 19-s − 6.20·21-s + 7.74·23-s − 2.43·25-s − 3.00·27-s + 5.34·29-s + 7.43·31-s + 2.14·33-s + 4.63·35-s − 7.49·37-s + 10.4·39-s − 2.05·41-s + 10.6·43-s − 2.56·45-s + 7.08·47-s + 1.36·49-s − 12.3·51-s + 8.32·53-s − 1.60·55-s + ⋯ |
L(s) = 1 | + 1.23·3-s − 0.716·5-s − 1.09·7-s + 0.533·9-s + 0.301·11-s + 1.35·13-s − 0.886·15-s − 1.39·17-s + 0.229·19-s − 1.35·21-s + 1.61·23-s − 0.487·25-s − 0.577·27-s + 0.993·29-s + 1.33·31-s + 0.373·33-s + 0.782·35-s − 1.23·37-s + 1.68·39-s − 0.321·41-s + 1.63·43-s − 0.382·45-s + 1.03·47-s + 0.194·49-s − 1.72·51-s + 1.14·53-s − 0.215·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.274335430\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.274335430\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 11 | \( 1 - T \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 - 2.14T + 3T^{2} \) |
| 5 | \( 1 + 1.60T + 5T^{2} \) |
| 7 | \( 1 + 2.89T + 7T^{2} \) |
| 13 | \( 1 - 4.89T + 13T^{2} \) |
| 17 | \( 1 + 5.74T + 17T^{2} \) |
| 23 | \( 1 - 7.74T + 23T^{2} \) |
| 29 | \( 1 - 5.34T + 29T^{2} \) |
| 31 | \( 1 - 7.43T + 31T^{2} \) |
| 37 | \( 1 + 7.49T + 37T^{2} \) |
| 41 | \( 1 + 2.05T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 7.08T + 47T^{2} \) |
| 53 | \( 1 - 8.32T + 53T^{2} \) |
| 59 | \( 1 - 2.54T + 59T^{2} \) |
| 61 | \( 1 - 13.4T + 61T^{2} \) |
| 67 | \( 1 - 10.3T + 67T^{2} \) |
| 71 | \( 1 + 14.9T + 71T^{2} \) |
| 73 | \( 1 + 3.74T + 73T^{2} \) |
| 79 | \( 1 - 8.69T + 79T^{2} \) |
| 83 | \( 1 - 6.10T + 83T^{2} \) |
| 89 | \( 1 - 3.20T + 89T^{2} \) |
| 97 | \( 1 + 6.98T + 97T^{2} \) |
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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
−8.744507718134895209416347988660, −8.114986168051229965878598529773, −7.07567106228648410087452310696, −6.65875021716116288090863418964, −5.71190213118998710536517377098, −4.41151625604023429027447590391, −3.75019801656087253981055009982, −3.12516300103931459250335527569, −2.33974577060600775093024291230, −0.845390758275827836033921617363,
0.845390758275827836033921617363, 2.33974577060600775093024291230, 3.12516300103931459250335527569, 3.75019801656087253981055009982, 4.41151625604023429027447590391, 5.71190213118998710536517377098, 6.65875021716116288090863418964, 7.07567106228648410087452310696, 8.114986168051229965878598529773, 8.744507718134895209416347988660