| L(s) = 1 | + 11.1·5-s − 18.3·7-s + 23.5·11-s − 67.7·13-s + 117.·17-s − 110.·19-s − 69.2·23-s + 0.353·25-s + 198.·29-s + 311.·31-s − 205.·35-s − 206.·37-s − 132.·41-s + 335.·43-s + 379.·47-s − 4.72·49-s + 190.·53-s + 263.·55-s + 337.·59-s + 277.·61-s − 758.·65-s − 665.·67-s + 528.·71-s − 73.8·73-s − 433.·77-s + 479.·79-s + 179.·83-s + ⋯ |
| L(s) = 1 | + 1.00·5-s − 0.993·7-s + 0.646·11-s − 1.44·13-s + 1.67·17-s − 1.33·19-s − 0.627·23-s + 0.00283·25-s + 1.27·29-s + 1.80·31-s − 0.994·35-s − 0.918·37-s − 0.505·41-s + 1.18·43-s + 1.17·47-s − 0.0137·49-s + 0.494·53-s + 0.646·55-s + 0.745·59-s + 0.582·61-s − 1.44·65-s − 1.21·67-s + 0.883·71-s − 0.118·73-s − 0.641·77-s + 0.683·79-s + 0.237·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1296 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.153719178\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.153719178\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 11.1T + 125T^{2} \) |
| 7 | \( 1 + 18.3T + 343T^{2} \) |
| 11 | \( 1 - 23.5T + 1.33e3T^{2} \) |
| 13 | \( 1 + 67.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 117.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 110.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 69.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 198.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 311.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 206.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 132.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 335.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 379.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 190.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 337.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 277.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 665.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 528.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 73.8T + 3.89e5T^{2} \) |
| 79 | \( 1 - 479.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 179.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 846.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 672.T + 9.12e5T^{2} \) |
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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
−9.489694953782319121947130090263, −8.598094026206431239904723260129, −7.59434615486001132768153171002, −6.60102991488792171111946212491, −6.10595920784799339355171312873, −5.17148690954281548284394110614, −4.10634010714314883570946477037, −2.94595861859702559137266807260, −2.10665245194355662178250806808, −0.72962201709322998593123671302,
0.72962201709322998593123671302, 2.10665245194355662178250806808, 2.94595861859702559137266807260, 4.10634010714314883570946477037, 5.17148690954281548284394110614, 6.10595920784799339355171312873, 6.60102991488792171111946212491, 7.59434615486001132768153171002, 8.598094026206431239904723260129, 9.489694953782319121947130090263
