| L(s) = 1 | − 3-s + 3.77·5-s − 7-s + 9-s + 1.77·11-s + 13-s − 3.77·15-s + 4.71·17-s + 2.71·19-s + 21-s + 2.71·23-s + 9.27·25-s − 27-s − 0.719·29-s − 1.88·31-s − 1.77·33-s − 3.77·35-s + 2.83·37-s − 39-s + 1.05·41-s − 0.837·43-s + 3.77·45-s − 10.3·47-s + 49-s − 4.71·51-s − 9.11·53-s + 6.71·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.68·5-s − 0.377·7-s + 0.333·9-s + 0.536·11-s + 0.277·13-s − 0.975·15-s + 1.14·17-s + 0.623·19-s + 0.218·21-s + 0.567·23-s + 1.85·25-s − 0.192·27-s − 0.133·29-s − 0.338·31-s − 0.309·33-s − 0.638·35-s + 0.466·37-s − 0.160·39-s + 0.165·41-s − 0.127·43-s + 0.563·45-s − 1.51·47-s + 0.142·49-s − 0.660·51-s − 1.25·53-s + 0.906·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.480105629\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.480105629\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 - 3.77T + 5T^{2} \) |
| 11 | \( 1 - 1.77T + 11T^{2} \) |
| 17 | \( 1 - 4.71T + 17T^{2} \) |
| 19 | \( 1 - 2.71T + 19T^{2} \) |
| 23 | \( 1 - 2.71T + 23T^{2} \) |
| 29 | \( 1 + 0.719T + 29T^{2} \) |
| 31 | \( 1 + 1.88T + 31T^{2} \) |
| 37 | \( 1 - 2.83T + 37T^{2} \) |
| 41 | \( 1 - 1.05T + 41T^{2} \) |
| 43 | \( 1 + 0.837T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 9.11T + 53T^{2} \) |
| 59 | \( 1 - 6.38T + 59T^{2} \) |
| 61 | \( 1 - 1.16T + 61T^{2} \) |
| 67 | \( 1 - 5.67T + 67T^{2} \) |
| 71 | \( 1 - 2.61T + 71T^{2} \) |
| 73 | \( 1 - 13.9T + 73T^{2} \) |
| 79 | \( 1 - 3.55T + 79T^{2} \) |
| 83 | \( 1 + 8.05T + 83T^{2} \) |
| 89 | \( 1 + 2.49T + 89T^{2} \) |
| 97 | \( 1 + 13.1T + 97T^{2} \) |
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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
−8.469149960006273778801352116433, −7.49500979906720801438777526745, −6.60712324579696724859702650450, −6.21569255436408621967818585072, −5.43153851781900050483570531011, −5.00424349778171147595303441524, −3.72839849365143803474148212821, −2.87020508153198669197296184961, −1.76246252404270836212503207960, −0.989497120956664713315628384540,
0.989497120956664713315628384540, 1.76246252404270836212503207960, 2.87020508153198669197296184961, 3.72839849365143803474148212821, 5.00424349778171147595303441524, 5.43153851781900050483570531011, 6.21569255436408621967818585072, 6.60712324579696724859702650450, 7.49500979906720801438777526745, 8.469149960006273778801352116433
