| L(s) = 1 | − 0.536·2-s − 1.71·4-s − 2.69·5-s − 1.96·7-s + 1.99·8-s + 1.44·10-s − 5.94·11-s + 3.20·13-s + 1.05·14-s + 2.35·16-s + 1.05·17-s + 2.01·19-s + 4.61·20-s + 3.19·22-s − 2.45·23-s + 2.25·25-s − 1.71·26-s + 3.36·28-s − 6.74·29-s + 0.586·31-s − 5.24·32-s − 0.568·34-s + 5.29·35-s − 10.8·37-s − 1.08·38-s − 5.36·40-s + 5.70·41-s + ⋯ |
| L(s) = 1 | − 0.379·2-s − 0.856·4-s − 1.20·5-s − 0.742·7-s + 0.704·8-s + 0.457·10-s − 1.79·11-s + 0.888·13-s + 0.281·14-s + 0.588·16-s + 0.256·17-s + 0.462·19-s + 1.03·20-s + 0.680·22-s − 0.511·23-s + 0.451·25-s − 0.336·26-s + 0.636·28-s − 1.25·29-s + 0.105·31-s − 0.927·32-s − 0.0974·34-s + 0.895·35-s − 1.78·37-s − 0.175·38-s − 0.848·40-s + 0.891·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6021 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.09497717983\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09497717983\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 223 | \( 1 - T \) |
| good | 2 | \( 1 + 0.536T + 2T^{2} \) |
| 5 | \( 1 + 2.69T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 + 5.94T + 11T^{2} \) |
| 13 | \( 1 - 3.20T + 13T^{2} \) |
| 17 | \( 1 - 1.05T + 17T^{2} \) |
| 19 | \( 1 - 2.01T + 19T^{2} \) |
| 23 | \( 1 + 2.45T + 23T^{2} \) |
| 29 | \( 1 + 6.74T + 29T^{2} \) |
| 31 | \( 1 - 0.586T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 5.70T + 41T^{2} \) |
| 43 | \( 1 + 7.50T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 + 11.9T + 53T^{2} \) |
| 59 | \( 1 + 5.42T + 59T^{2} \) |
| 61 | \( 1 - 7.38T + 61T^{2} \) |
| 67 | \( 1 + 9.91T + 67T^{2} \) |
| 71 | \( 1 - 2.51T + 71T^{2} \) |
| 73 | \( 1 + 7.15T + 73T^{2} \) |
| 79 | \( 1 - 4.84T + 79T^{2} \) |
| 83 | \( 1 - 5.53T + 83T^{2} \) |
| 89 | \( 1 + 5.29T + 89T^{2} \) |
| 97 | \( 1 + 18.4T + 97T^{2} \) |
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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
−8.098202198436019185368523772081, −7.66694528123536375467369459394, −6.88307503306596886374800662911, −5.80827254862056128374839202794, −5.17417084587234163825383125400, −4.40786832483549068976750422716, −3.49425319839114217167428275185, −3.16708048363308456558259662417, −1.63399051938886800519786825020, −0.17144036957277591137645783074,
0.17144036957277591137645783074, 1.63399051938886800519786825020, 3.16708048363308456558259662417, 3.49425319839114217167428275185, 4.40786832483549068976750422716, 5.17417084587234163825383125400, 5.80827254862056128374839202794, 6.88307503306596886374800662911, 7.66694528123536375467369459394, 8.098202198436019185368523772081
