| L(s) = 1 | − 0.588·2-s − 0.773·3-s − 1.65·4-s + 2.69·5-s + 0.455·6-s + 2.15·8-s − 2.40·9-s − 1.58·10-s − 2.69·11-s + 1.27·12-s − 2.08·15-s + 2.04·16-s + 0.475·17-s + 1.41·18-s − 1.96·19-s − 4.45·20-s + 1.58·22-s − 5.72·23-s − 1.66·24-s + 2.26·25-s + 4.17·27-s + 1.40·29-s + 1.22·30-s + 3.31·31-s − 5.50·32-s + 2.08·33-s − 0.279·34-s + ⋯ |
| L(s) = 1 | − 0.416·2-s − 0.446·3-s − 0.826·4-s + 1.20·5-s + 0.185·6-s + 0.760·8-s − 0.800·9-s − 0.501·10-s − 0.811·11-s + 0.369·12-s − 0.538·15-s + 0.510·16-s + 0.115·17-s + 0.333·18-s − 0.449·19-s − 0.996·20-s + 0.337·22-s − 1.19·23-s − 0.339·24-s + 0.453·25-s + 0.804·27-s + 0.261·29-s + 0.224·30-s + 0.594·31-s − 0.972·32-s + 0.362·33-s − 0.0479·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8281 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 0.588T + 2T^{2} \) |
| 3 | \( 1 + 0.773T + 3T^{2} \) |
| 5 | \( 1 - 2.69T + 5T^{2} \) |
| 11 | \( 1 + 2.69T + 11T^{2} \) |
| 17 | \( 1 - 0.475T + 17T^{2} \) |
| 19 | \( 1 + 1.96T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 - 1.40T + 29T^{2} \) |
| 31 | \( 1 - 3.31T + 31T^{2} \) |
| 37 | \( 1 - 10.8T + 37T^{2} \) |
| 41 | \( 1 - 11.6T + 41T^{2} \) |
| 43 | \( 1 + 9.43T + 43T^{2} \) |
| 47 | \( 1 - 8.98T + 47T^{2} \) |
| 53 | \( 1 + 4.30T + 53T^{2} \) |
| 59 | \( 1 + 9.92T + 59T^{2} \) |
| 61 | \( 1 - 9.96T + 61T^{2} \) |
| 67 | \( 1 + 3.40T + 67T^{2} \) |
| 71 | \( 1 - 5.27T + 71T^{2} \) |
| 73 | \( 1 - 4.21T + 73T^{2} \) |
| 79 | \( 1 + 14.7T + 79T^{2} \) |
| 83 | \( 1 + 4.99T + 83T^{2} \) |
| 89 | \( 1 + 3.67T + 89T^{2} \) |
| 97 | \( 1 - 17.3T + 97T^{2} \) |
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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
−7.79291434721623517344501645087, −6.57747129261258042050354456139, −5.89515231808619850551793791440, −5.57070085446663898427241599991, −4.76151086560772998793284769234, −4.08348439653761670153416144657, −2.85995648771063167741445295345, −2.18760467275803170075147712557, −1.05458178845350272882426174679, 0,
1.05458178845350272882426174679, 2.18760467275803170075147712557, 2.85995648771063167741445295345, 4.08348439653761670153416144657, 4.76151086560772998793284769234, 5.57070085446663898427241599991, 5.89515231808619850551793791440, 6.57747129261258042050354456139, 7.79291434721623517344501645087
