L(s) = 1 | − 2-s + 4-s + 3·5-s − 3·7-s − 8-s − 3·10-s + 3·14-s + 16-s + 3·17-s − 6·19-s + 3·20-s − 6·23-s + 4·25-s − 3·28-s − 32-s − 3·34-s − 9·35-s − 3·37-s + 6·38-s − 3·40-s + 43-s + 6·46-s + 3·47-s + 2·49-s − 4·50-s + 6·53-s + 3·56-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 1/2·4-s + 1.34·5-s − 1.13·7-s − 0.353·8-s − 0.948·10-s + 0.801·14-s + 1/4·16-s + 0.727·17-s − 1.37·19-s + 0.670·20-s − 1.25·23-s + 4/5·25-s − 0.566·28-s − 0.176·32-s − 0.514·34-s − 1.52·35-s − 0.493·37-s + 0.973·38-s − 0.474·40-s + 0.152·43-s + 0.884·46-s + 0.437·47-s + 2/7·49-s − 0.565·50-s + 0.824·53-s + 0.400·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3042 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 13 | \( 1 \) |
good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + 15 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−8.571423131733288125113369380721, −7.60233865517211300096655945141, −6.75973359536125424427821286498, −6.01357873304239857580488774330, −5.77655495511337207389488133517, −4.40034260373823956119588744066, −3.27914026857724435022736802635, −2.39071591840719444382300423431, −1.54439702007463877805293824731, 0,
1.54439702007463877805293824731, 2.39071591840719444382300423431, 3.27914026857724435022736802635, 4.40034260373823956119588744066, 5.77655495511337207389488133517, 6.01357873304239857580488774330, 6.75973359536125424427821286498, 7.60233865517211300096655945141, 8.571423131733288125113369380721