| L(s) = 1 | + 1.30·3-s + 2.30·5-s − 7-s − 1.30·9-s + 4·11-s + 2.60·13-s + 3·15-s − 17-s + 1.39·19-s − 1.30·21-s + 4·23-s + 0.302·25-s − 5.60·27-s − 5.21·29-s + 3.69·31-s + 5.21·33-s − 2.30·35-s − 11.8·37-s + 3.39·39-s + 6.51·41-s − 0.697·43-s − 3.00·45-s + 4.60·47-s + 49-s − 1.30·51-s + 4.30·53-s + 9.21·55-s + ⋯ |
| L(s) = 1 | + 0.752·3-s + 1.02·5-s − 0.377·7-s − 0.434·9-s + 1.20·11-s + 0.722·13-s + 0.774·15-s − 0.242·17-s + 0.319·19-s − 0.284·21-s + 0.834·23-s + 0.0605·25-s − 1.07·27-s − 0.967·29-s + 0.664·31-s + 0.907·33-s − 0.389·35-s − 1.94·37-s + 0.543·39-s + 1.01·41-s − 0.106·43-s − 0.447·45-s + 0.671·47-s + 0.142·49-s − 0.182·51-s + 0.591·53-s + 1.24·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 476 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.069244930\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.069244930\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 5 | \( 1 - 2.30T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 2.60T + 13T^{2} \) |
| 19 | \( 1 - 1.39T + 19T^{2} \) |
| 23 | \( 1 - 4T + 23T^{2} \) |
| 29 | \( 1 + 5.21T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 + 11.8T + 37T^{2} \) |
| 41 | \( 1 - 6.51T + 41T^{2} \) |
| 43 | \( 1 + 0.697T + 43T^{2} \) |
| 47 | \( 1 - 4.60T + 47T^{2} \) |
| 53 | \( 1 - 4.30T + 53T^{2} \) |
| 59 | \( 1 + 8T + 59T^{2} \) |
| 61 | \( 1 + 2.51T + 61T^{2} \) |
| 67 | \( 1 - 6.30T + 67T^{2} \) |
| 71 | \( 1 + 10.6T + 71T^{2} \) |
| 73 | \( 1 + 10.5T + 73T^{2} \) |
| 79 | \( 1 - 4.60T + 79T^{2} \) |
| 83 | \( 1 + 11.2T + 83T^{2} \) |
| 89 | \( 1 + 13.8T + 89T^{2} \) |
| 97 | \( 1 + 9.69T + 97T^{2} \) |
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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
−10.95314083862301695146980952542, −9.883127184923822987309935775602, −9.084510549202141237522329932422, −8.691166707003106484924568493522, −7.31021489287142883396217234915, −6.30036227906475194920740174462, −5.53163241973076577201627516367, −3.96049769465594348872907084961, −2.92304355581998684891020819472, −1.61734174675707997492615665344,
1.61734174675707997492615665344, 2.92304355581998684891020819472, 3.96049769465594348872907084961, 5.53163241973076577201627516367, 6.30036227906475194920740174462, 7.31021489287142883396217234915, 8.691166707003106484924568493522, 9.084510549202141237522329932422, 9.883127184923822987309935775602, 10.95314083862301695146980952542
