| L(s) = 1 | − 1.85·2-s + 0.655·3-s + 1.44·4-s + 4.10·5-s − 1.21·6-s − 1.24·7-s + 1.03·8-s − 2.57·9-s − 7.61·10-s − 11-s + 0.943·12-s + 2.04·13-s + 2.30·14-s + 2.69·15-s − 4.80·16-s − 17-s + 4.76·18-s + 5.72·19-s + 5.91·20-s − 0.814·21-s + 1.85·22-s − 5.52·23-s + 0.680·24-s + 11.8·25-s − 3.79·26-s − 3.65·27-s − 1.79·28-s + ⋯ |
| L(s) = 1 | − 1.31·2-s + 0.378·3-s + 0.720·4-s + 1.83·5-s − 0.496·6-s − 0.469·7-s + 0.366·8-s − 0.856·9-s − 2.40·10-s − 0.301·11-s + 0.272·12-s + 0.566·13-s + 0.616·14-s + 0.694·15-s − 1.20·16-s − 0.242·17-s + 1.12·18-s + 1.31·19-s + 1.32·20-s − 0.177·21-s + 0.395·22-s − 1.15·23-s + 0.138·24-s + 2.37·25-s − 0.743·26-s − 0.702·27-s − 0.338·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 + T \) |
| good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 3 | \( 1 - 0.655T + 3T^{2} \) |
| 5 | \( 1 - 4.10T + 5T^{2} \) |
| 7 | \( 1 + 1.24T + 7T^{2} \) |
| 13 | \( 1 - 2.04T + 13T^{2} \) |
| 19 | \( 1 - 5.72T + 19T^{2} \) |
| 23 | \( 1 + 5.52T + 23T^{2} \) |
| 29 | \( 1 + 8.74T + 29T^{2} \) |
| 31 | \( 1 - 2.93T + 31T^{2} \) |
| 37 | \( 1 - 1.56T + 37T^{2} \) |
| 41 | \( 1 + 6.25T + 41T^{2} \) |
| 47 | \( 1 + 7.49T + 47T^{2} \) |
| 53 | \( 1 - 1.86T + 53T^{2} \) |
| 59 | \( 1 + 8.57T + 59T^{2} \) |
| 61 | \( 1 - 6.86T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 7.96T + 71T^{2} \) |
| 73 | \( 1 + 12.0T + 73T^{2} \) |
| 79 | \( 1 - 6.30T + 79T^{2} \) |
| 83 | \( 1 + 7.80T + 83T^{2} \) |
| 89 | \( 1 - 2.05T + 89T^{2} \) |
| 97 | \( 1 + 6.35T + 97T^{2} \) |
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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
−7.73903222210747646559668530734, −6.85821451324877572187127892570, −6.17408437420805556557985968107, −5.61980816640652013864798954451, −4.92552462818203950290713039377, −3.59582887561260159296021001080, −2.75169703698135383918908412177, −1.99145753078116750514103494409, −1.33527495077386369091507554155, 0,
1.33527495077386369091507554155, 1.99145753078116750514103494409, 2.75169703698135383918908412177, 3.59582887561260159296021001080, 4.92552462818203950290713039377, 5.61980816640652013864798954451, 6.17408437420805556557985968107, 6.85821451324877572187127892570, 7.73903222210747646559668530734
