| L(s) = 1 | − 0.347·2-s − 1.87·4-s + 2.34·5-s − 1.87·7-s + 1.34·8-s − 0.815·10-s + 5.06·11-s + 4.71·13-s + 0.652·14-s + 3.29·16-s + 0.347·19-s − 4.41·20-s − 1.75·22-s + 1.77·23-s + 0.509·25-s − 1.63·26-s + 3.53·28-s − 2.22·29-s + 1.94·31-s − 3.83·32-s − 4.41·35-s − 6.17·37-s − 0.120·38-s + 3.16·40-s − 5.17·41-s − 1.47·43-s − 9.51·44-s + ⋯ |
| L(s) = 1 | − 0.245·2-s − 0.939·4-s + 1.04·5-s − 0.710·7-s + 0.476·8-s − 0.257·10-s + 1.52·11-s + 1.30·13-s + 0.174·14-s + 0.822·16-s + 0.0796·19-s − 0.986·20-s − 0.374·22-s + 0.369·23-s + 0.101·25-s − 0.321·26-s + 0.667·28-s − 0.413·29-s + 0.349·31-s − 0.678·32-s − 0.745·35-s − 1.01·37-s − 0.0195·38-s + 0.500·40-s − 0.807·41-s − 0.225·43-s − 1.43·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.693277298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.693277298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 0.347T + 2T^{2} \) |
| 5 | \( 1 - 2.34T + 5T^{2} \) |
| 7 | \( 1 + 1.87T + 7T^{2} \) |
| 11 | \( 1 - 5.06T + 11T^{2} \) |
| 13 | \( 1 - 4.71T + 13T^{2} \) |
| 19 | \( 1 - 0.347T + 19T^{2} \) |
| 23 | \( 1 - 1.77T + 23T^{2} \) |
| 29 | \( 1 + 2.22T + 29T^{2} \) |
| 31 | \( 1 - 1.94T + 31T^{2} \) |
| 37 | \( 1 + 6.17T + 37T^{2} \) |
| 41 | \( 1 + 5.17T + 41T^{2} \) |
| 43 | \( 1 + 1.47T + 43T^{2} \) |
| 47 | \( 1 - 8.53T + 47T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + 5.00T + 59T^{2} \) |
| 61 | \( 1 + 0.184T + 61T^{2} \) |
| 67 | \( 1 + 2.44T + 67T^{2} \) |
| 71 | \( 1 - 9.92T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 - 4.43T + 79T^{2} \) |
| 83 | \( 1 + 13.5T + 83T^{2} \) |
| 89 | \( 1 - 6.32T + 89T^{2} \) |
| 97 | \( 1 - 9.27T + 97T^{2} \) |
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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
−8.884177158495773562544016590014, −8.557701439654247176054857702624, −7.28633149295388522233611194354, −6.40032331231391626414204866406, −5.93144183328065538697859604819, −5.02261653889261857781576744566, −3.92075101519948699458396914868, −3.40595413766273607271571662829, −1.84833895415362699910712641583, −0.919719038910020694689343030328,
0.919719038910020694689343030328, 1.84833895415362699910712641583, 3.40595413766273607271571662829, 3.92075101519948699458396914868, 5.02261653889261857781576744566, 5.93144183328065538697859604819, 6.40032331231391626414204866406, 7.28633149295388522233611194354, 8.557701439654247176054857702624, 8.884177158495773562544016590014
