| L(s) = 1 | + 1.46·2-s + 1.61·3-s + 0.151·4-s − 0.466·5-s + 2.37·6-s − 2.71·8-s − 0.381·9-s − 0.684·10-s + 0.244·12-s − 1.58·13-s − 0.755·15-s − 4.27·16-s + 5.22·17-s − 0.560·18-s + 4.22·19-s − 0.0706·20-s − 1.80·23-s − 4.38·24-s − 4.78·25-s − 2.32·26-s − 5.47·27-s − 2.71·29-s − 1.10·30-s − 1.29·31-s − 0.854·32-s + 7.66·34-s − 0.0577·36-s + ⋯ |
| L(s) = 1 | + 1.03·2-s + 0.934·3-s + 0.0756·4-s − 0.208·5-s + 0.968·6-s − 0.958·8-s − 0.127·9-s − 0.216·10-s + 0.0706·12-s − 0.438·13-s − 0.194·15-s − 1.06·16-s + 1.26·17-s − 0.132·18-s + 0.968·19-s − 0.0157·20-s − 0.376·23-s − 0.895·24-s − 0.956·25-s − 0.455·26-s − 1.05·27-s − 0.504·29-s − 0.202·30-s − 0.232·31-s − 0.150·32-s + 1.31·34-s − 0.00963·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.46T + 2T^{2} \) |
| 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + 0.466T + 5T^{2} \) |
| 13 | \( 1 + 1.58T + 13T^{2} \) |
| 17 | \( 1 - 5.22T + 17T^{2} \) |
| 19 | \( 1 - 4.22T + 19T^{2} \) |
| 23 | \( 1 + 1.80T + 23T^{2} \) |
| 29 | \( 1 + 2.71T + 29T^{2} \) |
| 31 | \( 1 + 1.29T + 31T^{2} \) |
| 37 | \( 1 + 1.94T + 37T^{2} \) |
| 41 | \( 1 + 1.04T + 41T^{2} \) |
| 43 | \( 1 + 8.70T + 43T^{2} \) |
| 47 | \( 1 - 6.39T + 47T^{2} \) |
| 53 | \( 1 + 13.2T + 53T^{2} \) |
| 59 | \( 1 - 8.60T + 59T^{2} \) |
| 61 | \( 1 + 15.2T + 61T^{2} \) |
| 67 | \( 1 + 4.67T + 67T^{2} \) |
| 71 | \( 1 - 9.74T + 71T^{2} \) |
| 73 | \( 1 + 13.3T + 73T^{2} \) |
| 79 | \( 1 + 3.58T + 79T^{2} \) |
| 83 | \( 1 + 17.2T + 83T^{2} \) |
| 89 | \( 1 - 8.91T + 89T^{2} \) |
| 97 | \( 1 - 2.70T + 97T^{2} \) |
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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
−7.78333927397889241500288648383, −7.12235257688554563102872122129, −5.98892201783789711222998979549, −5.54582993106243079003999335572, −4.77970433267581975263668454193, −3.87258426841584348940132049219, −3.32704096255109637775555627800, −2.77312062167419797996171170113, −1.65933045937470579987646583180, 0,
1.65933045937470579987646583180, 2.77312062167419797996171170113, 3.32704096255109637775555627800, 3.87258426841584348940132049219, 4.77970433267581975263668454193, 5.54582993106243079003999335572, 5.98892201783789711222998979549, 7.12235257688554563102872122129, 7.78333927397889241500288648383
