| L(s) = 1 | + 0.381·2-s + 1.61·3-s − 1.85·4-s − 5-s + 0.618·6-s − 1.47·8-s − 0.381·9-s − 0.381·10-s − 3·12-s − 1.23·13-s − 1.61·15-s + 3.14·16-s + 3.09·17-s − 0.145·18-s − 1.76·19-s + 1.85·20-s + 5.09·23-s − 2.38·24-s − 4·25-s − 0.472·26-s − 5.47·27-s + 4.61·29-s − 0.618·30-s + 4.23·31-s + 4.14·32-s + 1.18·34-s + 0.708·36-s + ⋯ |
| L(s) = 1 | + 0.270·2-s + 0.934·3-s − 0.927·4-s − 0.447·5-s + 0.252·6-s − 0.520·8-s − 0.127·9-s − 0.120·10-s − 0.866·12-s − 0.342·13-s − 0.417·15-s + 0.786·16-s + 0.749·17-s − 0.0343·18-s − 0.404·19-s + 0.414·20-s + 1.06·23-s − 0.486·24-s − 0.800·25-s − 0.0925·26-s − 1.05·27-s + 0.857·29-s − 0.112·30-s + 0.760·31-s + 0.732·32-s + 0.202·34-s + 0.118·36-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 | \( 1 - 0.381T + 2T^{2} \) |
| 3 | \( 1 - 1.61T + 3T^{2} \) |
| 5 | \( 1 + T + 5T^{2} \) |
| 13 | \( 1 + 1.23T + 13T^{2} \) |
| 17 | \( 1 - 3.09T + 17T^{2} \) |
| 19 | \( 1 + 1.76T + 19T^{2} \) |
| 23 | \( 1 - 5.09T + 23T^{2} \) |
| 29 | \( 1 - 4.61T + 29T^{2} \) |
| 31 | \( 1 - 4.23T + 31T^{2} \) |
| 37 | \( 1 - 6.47T + 37T^{2} \) |
| 41 | \( 1 + 11.1T + 41T^{2} \) |
| 43 | \( 1 - 12.5T + 43T^{2} \) |
| 47 | \( 1 - 6.61T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 + 11.0T + 59T^{2} \) |
| 61 | \( 1 + 7.61T + 61T^{2} \) |
| 67 | \( 1 + 8.32T + 67T^{2} \) |
| 71 | \( 1 + 16.0T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 + 6.38T + 79T^{2} \) |
| 83 | \( 1 - 2.70T + 83T^{2} \) |
| 89 | \( 1 + 6.85T + 89T^{2} \) |
| 97 | \( 1 + 7T + 97T^{2} \) |
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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
−7.82526781839567359863816162251, −7.34211495869953952816037294596, −6.14827062255937091681800553319, −5.55692651692243480810799558171, −4.55315551983370695034741536627, −4.14410593776554375773748470577, −3.10460371812141705523158787158, −2.78187733711579423995896459641, −1.31345872150487694940412030624, 0,
1.31345872150487694940412030624, 2.78187733711579423995896459641, 3.10460371812141705523158787158, 4.14410593776554375773748470577, 4.55315551983370695034741536627, 5.55692651692243480810799558171, 6.14827062255937091681800553319, 7.34211495869953952816037294596, 7.82526781839567359863816162251
