| L(s) = 1 | + 2.56·3-s + 0.561·5-s − 0.561·7-s + 3.56·9-s + 2·11-s − 13-s + 1.43·15-s − 0.561·17-s + 6·19-s − 1.43·21-s − 4.68·25-s + 1.43·27-s − 8.24·29-s + 7.12·31-s + 5.12·33-s − 0.315·35-s − 9.68·37-s − 2.56·39-s + 7.12·41-s − 8.80·43-s + 2.00·45-s + 1.68·47-s − 6.68·49-s − 1.43·51-s − 4.87·53-s + 1.12·55-s + 15.3·57-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 0.251·5-s − 0.212·7-s + 1.18·9-s + 0.603·11-s − 0.277·13-s + 0.371·15-s − 0.136·17-s + 1.37·19-s − 0.313·21-s − 0.936·25-s + 0.276·27-s − 1.53·29-s + 1.27·31-s + 0.891·33-s − 0.0533·35-s − 1.59·37-s − 0.410·39-s + 1.11·41-s − 1.34·43-s + 0.298·45-s + 0.245·47-s − 0.954·49-s − 0.201·51-s − 0.669·53-s + 0.151·55-s + 2.03·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.238927647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.238927647\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.56T + 3T^{2} \) |
| 5 | \( 1 - 0.561T + 5T^{2} \) |
| 7 | \( 1 + 0.561T + 7T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 17 | \( 1 + 0.561T + 17T^{2} \) |
| 19 | \( 1 - 6T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 - 7.12T + 31T^{2} \) |
| 37 | \( 1 + 9.68T + 37T^{2} \) |
| 41 | \( 1 - 7.12T + 41T^{2} \) |
| 43 | \( 1 + 8.80T + 43T^{2} \) |
| 47 | \( 1 - 1.68T + 47T^{2} \) |
| 53 | \( 1 + 4.87T + 53T^{2} \) |
| 59 | \( 1 + 6T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 6T + 67T^{2} \) |
| 71 | \( 1 + 1.68T + 71T^{2} \) |
| 73 | \( 1 - 10T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 + 17.3T + 83T^{2} \) |
| 89 | \( 1 + 8.24T + 89T^{2} \) |
| 97 | \( 1 + 6T + 97T^{2} \) |
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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
−11.23461220324178498488048635367, −9.751879759476835518902712204197, −9.554414948318633638956058359001, −8.491922494420523496462340846268, −7.68638232220040779727430308657, −6.74646716726994014464223907671, −5.39667206829653893083510538697, −3.94563140609196492072449267321, −3.05843773369488483717877797781, −1.79757123529249810532635130140,
1.79757123529249810532635130140, 3.05843773369488483717877797781, 3.94563140609196492072449267321, 5.39667206829653893083510538697, 6.74646716726994014464223907671, 7.68638232220040779727430308657, 8.491922494420523496462340846268, 9.554414948318633638956058359001, 9.751879759476835518902712204197, 11.23461220324178498488048635367
