| L(s) = 1 | + 1.24·2-s + 3-s − 0.445·4-s − 2.80·5-s + 1.24·6-s − 4.80·7-s − 3.04·8-s + 9-s − 3.49·10-s + 1.46·11-s − 0.445·12-s − 5.98·14-s − 2.80·15-s − 2.91·16-s − 2.44·17-s + 1.24·18-s + 2.54·19-s + 1.24·20-s − 4.80·21-s + 1.82·22-s − 3.51·23-s − 3.04·24-s + 2.85·25-s + 27-s + 2.13·28-s + 1.85·29-s − 3.49·30-s + ⋯ |
| L(s) = 1 | + 0.881·2-s + 0.577·3-s − 0.222·4-s − 1.25·5-s + 0.509·6-s − 1.81·7-s − 1.07·8-s + 0.333·9-s − 1.10·10-s + 0.442·11-s − 0.128·12-s − 1.60·14-s − 0.723·15-s − 0.727·16-s − 0.593·17-s + 0.293·18-s + 0.583·19-s + 0.278·20-s − 1.04·21-s + 0.389·22-s − 0.733·23-s − 0.622·24-s + 0.570·25-s + 0.192·27-s + 0.403·28-s + 0.343·29-s − 0.637·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 1.24T + 2T^{2} \) |
| 5 | \( 1 + 2.80T + 5T^{2} \) |
| 7 | \( 1 + 4.80T + 7T^{2} \) |
| 11 | \( 1 - 1.46T + 11T^{2} \) |
| 17 | \( 1 + 2.44T + 17T^{2} \) |
| 19 | \( 1 - 2.54T + 19T^{2} \) |
| 23 | \( 1 + 3.51T + 23T^{2} \) |
| 29 | \( 1 - 1.85T + 29T^{2} \) |
| 31 | \( 1 + 7.63T + 31T^{2} \) |
| 37 | \( 1 + 4.55T + 37T^{2} \) |
| 41 | \( 1 - 1.24T + 41T^{2} \) |
| 43 | \( 1 - 2.38T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 8.85T + 53T^{2} \) |
| 59 | \( 1 - 2.17T + 59T^{2} \) |
| 61 | \( 1 + 7.82T + 61T^{2} \) |
| 67 | \( 1 + 3.58T + 67T^{2} \) |
| 71 | \( 1 + 8.83T + 71T^{2} \) |
| 73 | \( 1 + 7.69T + 73T^{2} \) |
| 79 | \( 1 + 4.02T + 79T^{2} \) |
| 83 | \( 1 + 0.652T + 83T^{2} \) |
| 89 | \( 1 - 6.29T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−10.44099667716575540984541779662, −9.328664358982393923814901433342, −8.901916956417484952978620653598, −7.62837194704805766727118585463, −6.73304836595896926602913362769, −5.75526842125667722901185388159, −4.26503047454121388669926092409, −3.69110187342332593466895818269, −2.91060722505024791174012062239, 0,
2.91060722505024791174012062239, 3.69110187342332593466895818269, 4.26503047454121388669926092409, 5.75526842125667722901185388159, 6.73304836595896926602913362769, 7.62837194704805766727118585463, 8.901916956417484952978620653598, 9.328664358982393923814901433342, 10.44099667716575540984541779662
