| L(s) = 1 | + 1.67·2-s + 0.792·4-s − 3.89·5-s + 4.51·7-s − 2.01·8-s − 6.51·10-s − 0.335·11-s − 1.74·13-s + 7.54·14-s − 4.95·16-s + 3.47·17-s + 0.211·19-s − 3.08·20-s − 0.560·22-s − 2.74·23-s + 10.1·25-s − 2.91·26-s + 3.57·28-s + 7.23·29-s − 4.24·32-s + 5.79·34-s − 17.5·35-s − 7.41·37-s + 0.354·38-s + 7.85·40-s − 6.51·41-s + 3.05·43-s + ⋯ |
| L(s) = 1 | + 1.18·2-s + 0.396·4-s − 1.74·5-s + 1.70·7-s − 0.713·8-s − 2.05·10-s − 0.101·11-s − 0.484·13-s + 2.01·14-s − 1.23·16-s + 0.841·17-s + 0.0486·19-s − 0.690·20-s − 0.119·22-s − 0.571·23-s + 2.03·25-s − 0.572·26-s + 0.676·28-s + 1.34·29-s − 0.751·32-s + 0.994·34-s − 2.97·35-s − 1.21·37-s + 0.0574·38-s + 1.24·40-s − 1.01·41-s + 0.466·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8649 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.559778411\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.559778411\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 31 | \( 1 \) |
| good | 2 | \( 1 - 1.67T + 2T^{2} \) |
| 5 | \( 1 + 3.89T + 5T^{2} \) |
| 7 | \( 1 - 4.51T + 7T^{2} \) |
| 11 | \( 1 + 0.335T + 11T^{2} \) |
| 13 | \( 1 + 1.74T + 13T^{2} \) |
| 17 | \( 1 - 3.47T + 17T^{2} \) |
| 19 | \( 1 - 0.211T + 19T^{2} \) |
| 23 | \( 1 + 2.74T + 23T^{2} \) |
| 29 | \( 1 - 7.23T + 29T^{2} \) |
| 37 | \( 1 + 7.41T + 37T^{2} \) |
| 41 | \( 1 + 6.51T + 41T^{2} \) |
| 43 | \( 1 - 3.05T + 43T^{2} \) |
| 47 | \( 1 + 3.33T + 47T^{2} \) |
| 53 | \( 1 + 12.1T + 53T^{2} \) |
| 59 | \( 1 - 1.37T + 59T^{2} \) |
| 61 | \( 1 - 3.58T + 61T^{2} \) |
| 67 | \( 1 + 1.37T + 67T^{2} \) |
| 71 | \( 1 - 1.73T + 71T^{2} \) |
| 73 | \( 1 - 13.0T + 73T^{2} \) |
| 79 | \( 1 - 8.50T + 79T^{2} \) |
| 83 | \( 1 - 7.95T + 83T^{2} \) |
| 89 | \( 1 - 8.05T + 89T^{2} \) |
| 97 | \( 1 + 3.44T + 97T^{2} \) |
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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
−7.82508340547894869698344982961, −7.13184078508846841241554296073, −6.32973985895105530782055096672, −5.17950123042331712510189382413, −4.95905942107635470472511728394, −4.34909810246325123646423682114, −3.65761209388290390549076086115, −3.05241945229474594116374426861, −1.92626482816974233687060936501, −0.64919586873917901558152155119,
0.64919586873917901558152155119, 1.92626482816974233687060936501, 3.05241945229474594116374426861, 3.65761209388290390549076086115, 4.34909810246325123646423682114, 4.95905942107635470472511728394, 5.17950123042331712510189382413, 6.32973985895105530782055096672, 7.13184078508846841241554296073, 7.82508340547894869698344982961
