| L(s) = 1 | − 3-s + 5-s − 5·7-s − 2·9-s + 5·11-s − 15-s − 17-s + 3·19-s + 5·21-s + 3·23-s + 25-s + 5·27-s − 29-s − 5·33-s − 5·35-s − 7·37-s + 5·41-s + 5·43-s − 2·45-s − 12·47-s + 18·49-s + 51-s + 2·53-s + 5·55-s − 3·57-s + 11·59-s − 13·61-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.88·7-s − 2/3·9-s + 1.50·11-s − 0.258·15-s − 0.242·17-s + 0.688·19-s + 1.09·21-s + 0.625·23-s + 1/5·25-s + 0.962·27-s − 0.185·29-s − 0.870·33-s − 0.845·35-s − 1.15·37-s + 0.780·41-s + 0.762·43-s − 0.298·45-s − 1.75·47-s + 18/7·49-s + 0.140·51-s + 0.274·53-s + 0.674·55-s − 0.397·57-s + 1.43·59-s − 1.66·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 5 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 11 T + p T^{2} \) |
| 61 | \( 1 + 13 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 11 T + p T^{2} \) |
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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
−8.460415525495013616829626697861, −7.04285104050544034963755324601, −6.74292467777373658486315014589, −5.98424744102755623520251490861, −5.51886781944076155191880539681, −4.30783005291139663875684693671, −3.38336742044400770919771396736, −2.76363414108881613872744126515, −1.26468429758175044176263363838, 0,
1.26468429758175044176263363838, 2.76363414108881613872744126515, 3.38336742044400770919771396736, 4.30783005291139663875684693671, 5.51886781944076155191880539681, 5.98424744102755623520251490861, 6.74292467777373658486315014589, 7.04285104050544034963755324601, 8.460415525495013616829626697861
