| L(s) = 1 | − 0.871·2-s + 3-s − 1.24·4-s − 4.06·5-s − 0.871·6-s + 7-s + 2.82·8-s + 9-s + 3.54·10-s − 1.24·12-s + 6.09·13-s − 0.871·14-s − 4.06·15-s + 0.0189·16-s − 1.70·17-s − 0.871·18-s − 5.45·19-s + 5.04·20-s + 21-s − 6.39·23-s + 2.82·24-s + 11.5·25-s − 5.30·26-s + 27-s − 1.24·28-s − 4.74·29-s + 3.54·30-s + ⋯ |
| L(s) = 1 | − 0.616·2-s + 0.577·3-s − 0.620·4-s − 1.81·5-s − 0.355·6-s + 0.377·7-s + 0.998·8-s + 0.333·9-s + 1.12·10-s − 0.358·12-s + 1.68·13-s − 0.232·14-s − 1.04·15-s + 0.00474·16-s − 0.412·17-s − 0.205·18-s − 1.25·19-s + 1.12·20-s + 0.218·21-s − 1.33·23-s + 0.576·24-s + 2.30·25-s − 1.04·26-s + 0.192·27-s − 0.234·28-s − 0.881·29-s + 0.646·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7988785581\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7988785581\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 0.871T + 2T^{2} \) |
| 5 | \( 1 + 4.06T + 5T^{2} \) |
| 13 | \( 1 - 6.09T + 13T^{2} \) |
| 17 | \( 1 + 1.70T + 17T^{2} \) |
| 19 | \( 1 + 5.45T + 19T^{2} \) |
| 23 | \( 1 + 6.39T + 23T^{2} \) |
| 29 | \( 1 + 4.74T + 29T^{2} \) |
| 31 | \( 1 + 4.36T + 31T^{2} \) |
| 37 | \( 1 - 2.43T + 37T^{2} \) |
| 41 | \( 1 - 3.21T + 41T^{2} \) |
| 43 | \( 1 + 0.127T + 43T^{2} \) |
| 47 | \( 1 - 8.46T + 47T^{2} \) |
| 53 | \( 1 + 4.71T + 53T^{2} \) |
| 59 | \( 1 - 4.63T + 59T^{2} \) |
| 61 | \( 1 + 5.31T + 61T^{2} \) |
| 67 | \( 1 - 14.0T + 67T^{2} \) |
| 71 | \( 1 - 2.21T + 71T^{2} \) |
| 73 | \( 1 - 5.57T + 73T^{2} \) |
| 79 | \( 1 + 6.27T + 79T^{2} \) |
| 83 | \( 1 - 0.127T + 83T^{2} \) |
| 89 | \( 1 + 8.12T + 89T^{2} \) |
| 97 | \( 1 + 2.59T + 97T^{2} \) |
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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
−8.647393918844743986968964823210, −8.260308923659139481749896028528, −7.76639232071939205566012194915, −6.97888806477405085141613915108, −5.84478536232217343308992595770, −4.51517348723367185249467209387, −4.02492534333679532165655782815, −3.53159595891986823532438810405, −1.92041113507822158862826689179, −0.61211689799365370865813705304,
0.61211689799365370865813705304, 1.92041113507822158862826689179, 3.53159595891986823532438810405, 4.02492534333679532165655782815, 4.51517348723367185249467209387, 5.84478536232217343308992595770, 6.97888806477405085141613915108, 7.76639232071939205566012194915, 8.260308923659139481749896028528, 8.647393918844743986968964823210
