| L(s) = 1 | + 3-s − 2.19·5-s + 7-s + 9-s − 5·11-s + 3.19·13-s − 2.19·15-s + 17-s + 3.38·19-s + 21-s − 23-s − 0.192·25-s + 27-s − 5·29-s + 3.38·31-s − 5·33-s − 2.19·35-s + 5.38·37-s + 3.19·39-s − 41-s − 5.19·43-s − 2.19·45-s − 10.3·47-s + 49-s + 51-s − 10.5·53-s + 10.9·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.980·5-s + 0.377·7-s + 0.333·9-s − 1.50·11-s + 0.885·13-s − 0.566·15-s + 0.242·17-s + 0.776·19-s + 0.218·21-s − 0.208·23-s − 0.0385·25-s + 0.192·27-s − 0.928·29-s + 0.607·31-s − 0.870·33-s − 0.370·35-s + 0.885·37-s + 0.511·39-s − 0.156·41-s − 0.791·43-s − 0.326·45-s − 1.51·47-s + 0.142·49-s + 0.140·51-s − 1.45·53-s + 1.47·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7728 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 + 2.19T + 5T^{2} \) |
| 11 | \( 1 + 5T + 11T^{2} \) |
| 13 | \( 1 - 3.19T + 13T^{2} \) |
| 17 | \( 1 - T + 17T^{2} \) |
| 19 | \( 1 - 3.38T + 19T^{2} \) |
| 29 | \( 1 + 5T + 29T^{2} \) |
| 31 | \( 1 - 3.38T + 31T^{2} \) |
| 37 | \( 1 - 5.38T + 37T^{2} \) |
| 41 | \( 1 + T + 41T^{2} \) |
| 43 | \( 1 + 5.19T + 43T^{2} \) |
| 47 | \( 1 + 10.3T + 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 9.19T + 61T^{2} \) |
| 67 | \( 1 + 2.19T + 67T^{2} \) |
| 71 | \( 1 + 13.5T + 71T^{2} \) |
| 73 | \( 1 + 9T + 73T^{2} \) |
| 79 | \( 1 - 9.77T + 79T^{2} \) |
| 83 | \( 1 - 11.3T + 83T^{2} \) |
| 89 | \( 1 - 3.57T + 89T^{2} \) |
| 97 | \( 1 + 17.7T + 97T^{2} \) |
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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
−7.77101008325541605634216473251, −7.08573969509191910202644796491, −6.11557847814457826750481680418, −5.29537235119669112638215405830, −4.65614016252587385015870264288, −3.76431188425055562228939000717, −3.23039397008500129541672834323, −2.36118927780005340615701164414, −1.28485741823958006760553428683, 0,
1.28485741823958006760553428683, 2.36118927780005340615701164414, 3.23039397008500129541672834323, 3.76431188425055562228939000717, 4.65614016252587385015870264288, 5.29537235119669112638215405830, 6.11557847814457826750481680418, 7.08573969509191910202644796491, 7.77101008325541605634216473251
