| L(s) = 1 | + 3-s − 7-s + 9-s − 4·11-s + 4·13-s − 6·17-s − 6·19-s − 21-s + 23-s + 27-s − 6·29-s + 2·31-s − 4·33-s − 2·37-s + 4·39-s + 2·41-s + 43-s − 47-s − 6·49-s − 6·51-s + 6·53-s − 6·57-s − 8·59-s + 10·61-s − 63-s − 7·67-s + 69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s − 1.45·17-s − 1.37·19-s − 0.218·21-s + 0.208·23-s + 0.192·27-s − 1.11·29-s + 0.359·31-s − 0.696·33-s − 0.328·37-s + 0.640·39-s + 0.312·41-s + 0.152·43-s − 0.145·47-s − 6/7·49-s − 0.840·51-s + 0.824·53-s − 0.794·57-s − 1.04·59-s + 1.28·61-s − 0.125·63-s − 0.855·67-s + 0.120·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 206400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 7 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−13.38131330107697, −12.86392058230770, −12.65234825855551, −11.83782484372682, −11.18367622993251, −10.99403160401450, −10.37363205554764, −10.15580007974990, −9.356473088063963, −8.821782406013957, −8.748222302557089, −8.036206951054243, −7.686972504845773, −7.024089628044554, −6.532262525426418, −6.122466474168617, −5.566994586422100, −4.868717018887476, −4.376068935678432, −3.893306357889120, −3.279522328040067, −2.758374854318243, −2.079201141858958, −1.797558426295911, −0.6984823041307621, 0,
0.6984823041307621, 1.797558426295911, 2.079201141858958, 2.758374854318243, 3.279522328040067, 3.893306357889120, 4.376068935678432, 4.868717018887476, 5.566994586422100, 6.122466474168617, 6.532262525426418, 7.024089628044554, 7.686972504845773, 8.036206951054243, 8.748222302557089, 8.821782406013957, 9.356473088063963, 10.15580007974990, 10.37363205554764, 10.99403160401450, 11.18367622993251, 11.83782484372682, 12.65234825855551, 12.86392058230770, 13.38131330107697
