| L(s) = 1 | + 1.19·3-s − 0.00829·5-s − 3.59·7-s − 1.57·9-s + 0.236·11-s + 1.29·13-s − 0.00988·15-s − 1.05·17-s − 2.02·19-s − 4.29·21-s − 0.342·23-s − 4.99·25-s − 5.45·27-s + 4.97·29-s − 4.91·31-s + 0.281·33-s + 0.0298·35-s − 10.6·37-s + 1.54·39-s + 8.34·41-s + 1.40·43-s + 0.0130·45-s + 0.719·47-s + 5.95·49-s − 1.25·51-s + 12.5·53-s − 0.00195·55-s + ⋯ |
| L(s) = 1 | + 0.688·3-s − 0.00370·5-s − 1.36·7-s − 0.526·9-s + 0.0712·11-s + 0.358·13-s − 0.00255·15-s − 0.255·17-s − 0.465·19-s − 0.936·21-s − 0.0713·23-s − 0.999·25-s − 1.05·27-s + 0.923·29-s − 0.883·31-s + 0.0490·33-s + 0.00504·35-s − 1.74·37-s + 0.246·39-s + 1.30·41-s + 0.214·43-s + 0.00195·45-s + 0.104·47-s + 0.851·49-s − 0.175·51-s + 1.72·53-s − 0.000264·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.492081080\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.492081080\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 1.19T + 3T^{2} \) |
| 5 | \( 1 + 0.00829T + 5T^{2} \) |
| 7 | \( 1 + 3.59T + 7T^{2} \) |
| 11 | \( 1 - 0.236T + 11T^{2} \) |
| 13 | \( 1 - 1.29T + 13T^{2} \) |
| 17 | \( 1 + 1.05T + 17T^{2} \) |
| 19 | \( 1 + 2.02T + 19T^{2} \) |
| 23 | \( 1 + 0.342T + 23T^{2} \) |
| 29 | \( 1 - 4.97T + 29T^{2} \) |
| 31 | \( 1 + 4.91T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 - 8.34T + 41T^{2} \) |
| 43 | \( 1 - 1.40T + 43T^{2} \) |
| 47 | \( 1 - 0.719T + 47T^{2} \) |
| 53 | \( 1 - 12.5T + 53T^{2} \) |
| 59 | \( 1 - 1.26T + 59T^{2} \) |
| 61 | \( 1 - 9.37T + 61T^{2} \) |
| 67 | \( 1 - 0.639T + 67T^{2} \) |
| 71 | \( 1 - 16.7T + 71T^{2} \) |
| 73 | \( 1 + 0.333T + 73T^{2} \) |
| 79 | \( 1 + 0.976T + 79T^{2} \) |
| 83 | \( 1 - 8.02T + 83T^{2} \) |
| 89 | \( 1 - 14.7T + 89T^{2} \) |
| 97 | \( 1 - 4.97T + 97T^{2} \) |
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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
−7.896289988685248294138667509985, −7.12175254203785583700954262960, −6.44983989352901100224619727293, −5.88002928621599390198423832726, −5.12133291221910426274315318349, −3.80912945377570837729785976335, −3.67196929478932340882552256536, −2.66405182805938770373663362398, −2.05946532645986293185180862991, −0.55025284238157093668544659261,
0.55025284238157093668544659261, 2.05946532645986293185180862991, 2.66405182805938770373663362398, 3.67196929478932340882552256536, 3.80912945377570837729785976335, 5.12133291221910426274315318349, 5.88002928621599390198423832726, 6.44983989352901100224619727293, 7.12175254203785583700954262960, 7.896289988685248294138667509985
