| L(s) = 1 | + 1.50·3-s − 4.06·5-s + 7-s − 0.729·9-s + 2.88·11-s − 1.77·13-s − 6.12·15-s + 5.57·17-s − 19-s + 1.50·21-s − 2.10·23-s + 11.5·25-s − 5.61·27-s − 0.313·29-s − 3.57·31-s + 4.35·33-s − 4.06·35-s − 4.55·37-s − 2.67·39-s + 0.953·41-s + 1.11·43-s + 2.96·45-s + 0.222·47-s + 49-s + 8.39·51-s − 0.0165·53-s − 11.7·55-s + ⋯ |
| L(s) = 1 | + 0.869·3-s − 1.81·5-s + 0.377·7-s − 0.243·9-s + 0.871·11-s − 0.492·13-s − 1.58·15-s + 1.35·17-s − 0.229·19-s + 0.328·21-s − 0.439·23-s + 2.30·25-s − 1.08·27-s − 0.0581·29-s − 0.641·31-s + 0.757·33-s − 0.687·35-s − 0.748·37-s − 0.428·39-s + 0.148·41-s + 0.170·43-s + 0.442·45-s + 0.0324·47-s + 0.142·49-s + 1.17·51-s − 0.00227·53-s − 1.58·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 1.50T + 3T^{2} \) |
| 5 | \( 1 + 4.06T + 5T^{2} \) |
| 11 | \( 1 - 2.88T + 11T^{2} \) |
| 13 | \( 1 + 1.77T + 13T^{2} \) |
| 17 | \( 1 - 5.57T + 17T^{2} \) |
| 23 | \( 1 + 2.10T + 23T^{2} \) |
| 29 | \( 1 + 0.313T + 29T^{2} \) |
| 31 | \( 1 + 3.57T + 31T^{2} \) |
| 37 | \( 1 + 4.55T + 37T^{2} \) |
| 41 | \( 1 - 0.953T + 41T^{2} \) |
| 43 | \( 1 - 1.11T + 43T^{2} \) |
| 47 | \( 1 - 0.222T + 47T^{2} \) |
| 53 | \( 1 + 0.0165T + 53T^{2} \) |
| 59 | \( 1 + 7.17T + 59T^{2} \) |
| 61 | \( 1 - 4.72T + 61T^{2} \) |
| 67 | \( 1 + 6.30T + 67T^{2} \) |
| 71 | \( 1 + 10.3T + 71T^{2} \) |
| 73 | \( 1 + 10.4T + 73T^{2} \) |
| 79 | \( 1 - 15.7T + 79T^{2} \) |
| 83 | \( 1 + 4.28T + 83T^{2} \) |
| 89 | \( 1 - 3.10T + 89T^{2} \) |
| 97 | \( 1 + 11.4T + 97T^{2} \) |
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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
−7.88248909302881877783525777576, −7.64560403378897906647454497144, −6.88143823588785230417491347157, −5.77236528390168296783292774505, −4.81855297972979761990704444045, −3.94780156920766219464587766136, −3.52707838178148786826351647129, −2.70372936397620374212788550598, −1.39424253169730478533561005341, 0,
1.39424253169730478533561005341, 2.70372936397620374212788550598, 3.52707838178148786826351647129, 3.94780156920766219464587766136, 4.81855297972979761990704444045, 5.77236528390168296783292774505, 6.88143823588785230417491347157, 7.64560403378897906647454497144, 7.88248909302881877783525777576
