| L(s) = 1 | − 1.50·2-s + 0.252·4-s + 0.569·5-s − 3.97·7-s + 2.62·8-s − 0.855·10-s − 11-s − 3.48·13-s + 5.96·14-s − 4.44·16-s − 7.75·17-s + 1.56·19-s + 0.143·20-s + 1.50·22-s − 4.91·23-s − 4.67·25-s + 5.23·26-s − 1.00·28-s − 6.29·29-s + 2.87·31-s + 1.41·32-s + 11.6·34-s − 2.26·35-s − 9.34·37-s − 2.34·38-s + 1.49·40-s − 1.58·41-s + ⋯ |
| L(s) = 1 | − 1.06·2-s + 0.126·4-s + 0.254·5-s − 1.50·7-s + 0.927·8-s − 0.270·10-s − 0.301·11-s − 0.967·13-s + 1.59·14-s − 1.11·16-s − 1.88·17-s + 0.357·19-s + 0.0321·20-s + 0.319·22-s − 1.02·23-s − 0.935·25-s + 1.02·26-s − 0.189·28-s − 1.16·29-s + 0.516·31-s + 0.250·32-s + 1.99·34-s − 0.382·35-s − 1.53·37-s − 0.379·38-s + 0.236·40-s − 0.247·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.05282325951\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.05282325951\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.50T + 2T^{2} \) |
| 5 | \( 1 - 0.569T + 5T^{2} \) |
| 7 | \( 1 + 3.97T + 7T^{2} \) |
| 13 | \( 1 + 3.48T + 13T^{2} \) |
| 17 | \( 1 + 7.75T + 17T^{2} \) |
| 19 | \( 1 - 1.56T + 19T^{2} \) |
| 23 | \( 1 + 4.91T + 23T^{2} \) |
| 29 | \( 1 + 6.29T + 29T^{2} \) |
| 31 | \( 1 - 2.87T + 31T^{2} \) |
| 37 | \( 1 + 9.34T + 37T^{2} \) |
| 41 | \( 1 + 1.58T + 41T^{2} \) |
| 43 | \( 1 - 5.31T + 43T^{2} \) |
| 47 | \( 1 + 3.46T + 47T^{2} \) |
| 53 | \( 1 + 10.8T + 53T^{2} \) |
| 59 | \( 1 + 0.418T + 59T^{2} \) |
| 67 | \( 1 - 9.84T + 67T^{2} \) |
| 71 | \( 1 + 9.93T + 71T^{2} \) |
| 73 | \( 1 - 7.11T + 73T^{2} \) |
| 79 | \( 1 + 13.1T + 79T^{2} \) |
| 83 | \( 1 + 4.04T + 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 + 1.77T + 97T^{2} \) |
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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
−8.120673128324600300123503064302, −7.45648463985113840739682262908, −6.79669467804188882929042412721, −6.19733733879651175601385555773, −5.25771454834155621807961662635, −4.39410113597000615451264521439, −3.60207029141601916212570425013, −2.50348011103919694704074360061, −1.80919977223428335133748038831, −0.13395458072333980008735686771,
0.13395458072333980008735686771, 1.80919977223428335133748038831, 2.50348011103919694704074360061, 3.60207029141601916212570425013, 4.39410113597000615451264521439, 5.25771454834155621807961662635, 6.19733733879651175601385555773, 6.79669467804188882929042412721, 7.45648463985113840739682262908, 8.120673128324600300123503064302
