| L(s) = 1 | − 2.82·3-s − 2·5-s + 4.24·7-s + 5.00·9-s − 1.41·11-s + 13-s + 5.65·15-s − 4·17-s − 1.41·19-s − 12·21-s − 5.65·23-s − 25-s − 5.65·27-s + 2·29-s + 9.89·31-s + 4.00·33-s − 8.48·35-s − 2·37-s − 2.82·39-s − 6·41-s + 11.3·43-s − 10.0·45-s + 7.07·47-s + 10.9·49-s + 11.3·51-s − 12·53-s + 2.82·55-s + ⋯ |
| L(s) = 1 | − 1.63·3-s − 0.894·5-s + 1.60·7-s + 1.66·9-s − 0.426·11-s + 0.277·13-s + 1.46·15-s − 0.970·17-s − 0.324·19-s − 2.61·21-s − 1.17·23-s − 0.200·25-s − 1.08·27-s + 0.371·29-s + 1.77·31-s + 0.696·33-s − 1.43·35-s − 0.328·37-s − 0.452·39-s − 0.937·41-s + 1.72·43-s − 1.49·45-s + 1.03·47-s + 1.57·49-s + 1.58·51-s − 1.64·53-s + 0.381·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 2.82T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 4.24T + 7T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 1.41T + 19T^{2} \) |
| 23 | \( 1 + 5.65T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 - 9.89T + 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 11.3T + 43T^{2} \) |
| 47 | \( 1 - 7.07T + 47T^{2} \) |
| 53 | \( 1 + 12T + 53T^{2} \) |
| 59 | \( 1 + 9.89T + 59T^{2} \) |
| 61 | \( 1 - 12T + 61T^{2} \) |
| 67 | \( 1 - 9.89T + 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + 14T + 73T^{2} \) |
| 79 | \( 1 + 8.48T + 79T^{2} \) |
| 83 | \( 1 - 7.07T + 83T^{2} \) |
| 89 | \( 1 - 14T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−8.059653507528078374872501750387, −7.59862685743149354946430271384, −6.61281844000940442731979600692, −5.97645560622952710799095694615, −5.08639022618836546178612427138, −4.53862362131279549401030651482, −3.99627030263469115923852930621, −2.31103028361210237895785709517, −1.16491194269421804000014717364, 0,
1.16491194269421804000014717364, 2.31103028361210237895785709517, 3.99627030263469115923852930621, 4.53862362131279549401030651482, 5.08639022618836546178612427138, 5.97645560622952710799095694615, 6.61281844000940442731979600692, 7.59862685743149354946430271384, 8.059653507528078374872501750387
