| L(s) = 1 | − 4·7-s − 13-s − 5·19-s − 3·29-s + 4·31-s + 7·37-s − 3·41-s + 2·43-s − 9·47-s + 9·49-s + 9·53-s + 6·59-s + 8·61-s + 5·67-s − 3·71-s + 4·73-s − 11·79-s + 6·83-s − 6·89-s + 4·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 0.277·13-s − 1.14·19-s − 0.557·29-s + 0.718·31-s + 1.15·37-s − 0.468·41-s + 0.304·43-s − 1.31·47-s + 9/7·49-s + 1.23·53-s + 0.781·59-s + 1.02·61-s + 0.610·67-s − 0.356·71-s + 0.468·73-s − 1.23·79-s + 0.658·83-s − 0.635·89-s + 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 46800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 7 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 9 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−14.94480932189723, −14.41109966102428, −13.68929791795053, −13.19101297847445, −12.87467320084595, −12.43972881797652, −11.77302830932168, −11.28808453813124, −10.59889327288989, −10.10040185053372, −9.652002833451386, −9.260280138147368, −8.470328842861699, −8.141231800466415, −7.272861487018973, −6.729035236527830, −6.430168495017939, −5.733192176795932, −5.207860252309029, −4.230286513598577, −3.962978592467677, −3.081278787771592, −2.631884671118120, −1.880503850261917, −0.7787349513776047, 0,
0.7787349513776047, 1.880503850261917, 2.631884671118120, 3.081278787771592, 3.962978592467677, 4.230286513598577, 5.207860252309029, 5.733192176795932, 6.430168495017939, 6.729035236527830, 7.272861487018973, 8.141231800466415, 8.470328842861699, 9.260280138147368, 9.652002833451386, 10.10040185053372, 10.59889327288989, 11.28808453813124, 11.77302830932168, 12.43972881797652, 12.87467320084595, 13.19101297847445, 13.68929791795053, 14.41109966102428, 14.94480932189723
