| L(s) = 1 | − 2·3-s − 5·7-s + 9-s + 4·11-s + 3·13-s + 6·17-s − 5·19-s + 10·21-s − 4·23-s + 4·27-s − 6·29-s − 4·31-s − 8·33-s + 8·37-s − 6·39-s − 3·41-s + 12·43-s − 2·47-s + 18·49-s − 12·51-s + 6·53-s + 10·57-s − 3·59-s − 8·61-s − 5·63-s − 5·67-s + 8·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.88·7-s + 1/3·9-s + 1.20·11-s + 0.832·13-s + 1.45·17-s − 1.14·19-s + 2.18·21-s − 0.834·23-s + 0.769·27-s − 1.11·29-s − 0.718·31-s − 1.39·33-s + 1.31·37-s − 0.960·39-s − 0.468·41-s + 1.82·43-s − 0.291·47-s + 18/7·49-s − 1.68·51-s + 0.824·53-s + 1.32·57-s − 0.390·59-s − 1.02·61-s − 0.629·63-s − 0.610·67-s + 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 155600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 155600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 389 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 13 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 8 T + p T^{2} \) |
| 97 | \( 1 - 9 T + p T^{2} \) |
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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
−13.53962231653488, −12.71856539405849, −12.62129802884839, −12.22042156827642, −11.67714098796374, −11.15863530460028, −10.73270001094615, −10.24893138677714, −9.749750810728657, −9.227796327425870, −9.022102786187348, −8.192713726598056, −7.626045047114356, −6.944286300885581, −6.536871912351999, −6.086839345104294, −5.831986140073081, −5.443112817422868, −4.414835150794509, −3.922845872361190, −3.590778165048865, −2.953266349203181, −2.180173516846586, −1.278931978891059, −0.6890020832187482, 0,
0.6890020832187482, 1.278931978891059, 2.180173516846586, 2.953266349203181, 3.590778165048865, 3.922845872361190, 4.414835150794509, 5.443112817422868, 5.831986140073081, 6.086839345104294, 6.536871912351999, 6.944286300885581, 7.626045047114356, 8.192713726598056, 9.022102786187348, 9.227796327425870, 9.749750810728657, 10.24893138677714, 10.73270001094615, 11.15863530460028, 11.67714098796374, 12.22042156827642, 12.62129802884839, 12.71856539405849, 13.53962231653488
