| L(s) = 1 | − 3-s − 5-s − 1.09·7-s + 9-s + 5.25·11-s + 1.24·13-s + 15-s − 2.64·17-s + 3.25·19-s + 1.09·21-s − 6.99·23-s + 25-s − 27-s + 5.89·29-s + 31-s − 5.25·33-s + 1.09·35-s − 3.24·37-s − 1.24·39-s + 4·41-s − 4.34·43-s − 45-s + 7.14·47-s − 5.80·49-s + 2.64·51-s + 6.80·53-s − 5.25·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 0.413·7-s + 0.333·9-s + 1.58·11-s + 0.346·13-s + 0.258·15-s − 0.641·17-s + 0.747·19-s + 0.238·21-s − 1.45·23-s + 0.200·25-s − 0.192·27-s + 1.09·29-s + 0.179·31-s − 0.915·33-s + 0.185·35-s − 0.534·37-s − 0.200·39-s + 0.624·41-s − 0.662·43-s − 0.149·45-s + 1.04·47-s − 0.828·49-s + 0.370·51-s + 0.934·53-s − 0.708·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.359398291\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.359398291\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 7 | \( 1 + 1.09T + 7T^{2} \) |
| 11 | \( 1 - 5.25T + 11T^{2} \) |
| 13 | \( 1 - 1.24T + 13T^{2} \) |
| 17 | \( 1 + 2.64T + 17T^{2} \) |
| 19 | \( 1 - 3.25T + 19T^{2} \) |
| 23 | \( 1 + 6.99T + 23T^{2} \) |
| 29 | \( 1 - 5.89T + 29T^{2} \) |
| 37 | \( 1 + 3.24T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 + 4.34T + 43T^{2} \) |
| 47 | \( 1 - 7.14T + 47T^{2} \) |
| 53 | \( 1 - 6.80T + 53T^{2} \) |
| 59 | \( 1 + 8.80T + 59T^{2} \) |
| 61 | \( 1 - 2.91T + 61T^{2} \) |
| 67 | \( 1 - 1.43T + 67T^{2} \) |
| 71 | \( 1 + 4.24T + 71T^{2} \) |
| 73 | \( 1 + 8.69T + 73T^{2} \) |
| 79 | \( 1 - 12.0T + 79T^{2} \) |
| 83 | \( 1 - 0.0433T + 83T^{2} \) |
| 89 | \( 1 - 4.63T + 89T^{2} \) |
| 97 | \( 1 - 0.309T + 97T^{2} \) |
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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
−8.582700506809040013183590541459, −7.72428913503810348554491418798, −6.84507741587962547329411037434, −6.38449272942826358006487972263, −5.67926337164662786597921635712, −4.55895239649700015890287367839, −4.00025442006870157291092781782, −3.17552935359166450437443391236, −1.81506559997777465915650910921, −0.71666207466301013753571570161,
0.71666207466301013753571570161, 1.81506559997777465915650910921, 3.17552935359166450437443391236, 4.00025442006870157291092781782, 4.55895239649700015890287367839, 5.67926337164662786597921635712, 6.38449272942826358006487972263, 6.84507741587962547329411037434, 7.72428913503810348554491418798, 8.582700506809040013183590541459
