| L(s) = 1 | − 0.899·3-s + 4.33·5-s + 7-s − 2.19·9-s + 4.31·11-s − 3.89·15-s + 4.97·17-s + 5.30·19-s − 0.899·21-s − 8.17·23-s + 13.7·25-s + 4.66·27-s + 5.38·29-s + 7.28·31-s − 3.87·33-s + 4.33·35-s − 8.60·37-s + 0.737·41-s + 5.04·43-s − 9.49·45-s − 4.91·47-s + 49-s − 4.47·51-s + 8.33·53-s + 18.6·55-s − 4.77·57-s − 3.81·59-s + ⋯ |
| L(s) = 1 | − 0.519·3-s + 1.93·5-s + 0.377·7-s − 0.730·9-s + 1.29·11-s − 1.00·15-s + 1.20·17-s + 1.21·19-s − 0.196·21-s − 1.70·23-s + 2.75·25-s + 0.898·27-s + 1.00·29-s + 1.30·31-s − 0.674·33-s + 0.732·35-s − 1.41·37-s + 0.115·41-s + 0.770·43-s − 1.41·45-s − 0.716·47-s + 0.142·49-s − 0.626·51-s + 1.14·53-s + 2.51·55-s − 0.632·57-s − 0.496·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.265029820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.265029820\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + 0.899T + 3T^{2} \) |
| 5 | \( 1 - 4.33T + 5T^{2} \) |
| 11 | \( 1 - 4.31T + 11T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 19 | \( 1 - 5.30T + 19T^{2} \) |
| 23 | \( 1 + 8.17T + 23T^{2} \) |
| 29 | \( 1 - 5.38T + 29T^{2} \) |
| 31 | \( 1 - 7.28T + 31T^{2} \) |
| 37 | \( 1 + 8.60T + 37T^{2} \) |
| 41 | \( 1 - 0.737T + 41T^{2} \) |
| 43 | \( 1 - 5.04T + 43T^{2} \) |
| 47 | \( 1 + 4.91T + 47T^{2} \) |
| 53 | \( 1 - 8.33T + 53T^{2} \) |
| 59 | \( 1 + 3.81T + 59T^{2} \) |
| 61 | \( 1 + 2.65T + 61T^{2} \) |
| 67 | \( 1 + 0.367T + 67T^{2} \) |
| 71 | \( 1 + 4.07T + 71T^{2} \) |
| 73 | \( 1 - 8.06T + 73T^{2} \) |
| 79 | \( 1 + 5.70T + 79T^{2} \) |
| 83 | \( 1 + 7.66T + 83T^{2} \) |
| 89 | \( 1 + 7.72T + 89T^{2} \) |
| 97 | \( 1 - 17.0T + 97T^{2} \) |
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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
−7.68614838002689123500906035359, −6.61457760445757993085161014003, −6.33426293830280806191097823355, −5.55020450295995762324701428826, −5.33983614036279544627991343491, −4.37021563985958792790687068210, −3.28184341751466776377769687792, −2.53546642558634150101316485596, −1.55766696634653400957399294996, −0.995304126748011374542753337015,
0.995304126748011374542753337015, 1.55766696634653400957399294996, 2.53546642558634150101316485596, 3.28184341751466776377769687792, 4.37021563985958792790687068210, 5.33983614036279544627991343491, 5.55020450295995762324701428826, 6.33426293830280806191097823355, 6.61457760445757993085161014003, 7.68614838002689123500906035359
