| L(s) = 1 | + 3-s − 5-s + 4·7-s + 9-s − 4·11-s − 15-s − 8·17-s + 4·21-s + 6·23-s + 25-s + 27-s + 8·29-s + 2·31-s − 4·33-s − 4·35-s + 4·37-s − 10·41-s − 4·43-s − 45-s − 8·47-s + 9·49-s − 8·51-s + 10·53-s + 4·55-s − 12·59-s + 2·61-s + 4·63-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.258·15-s − 1.94·17-s + 0.872·21-s + 1.25·23-s + 1/5·25-s + 0.192·27-s + 1.48·29-s + 0.359·31-s − 0.696·33-s − 0.676·35-s + 0.657·37-s − 1.56·41-s − 0.609·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s − 1.12·51-s + 1.37·53-s + 0.539·55-s − 1.56·59-s + 0.256·61-s + 0.503·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.57234811854798, −13.22224512103598, −12.59620836225219, −11.99523284199566, −11.57865715952723, −11.05683672654047, −10.74234370632051, −10.31441789601497, −9.664890648726984, −8.978307603905387, −8.480131859412686, −8.338852386295085, −7.850080526128699, −7.235697871095177, −6.817340406785334, −6.296159852389740, −5.335735599914506, −4.964138713872202, −4.531966204186097, −4.227976161858500, −3.202739104159290, −2.835719355561519, −2.185346738127398, −1.673317930679823, −0.8927644644715885, 0,
0.8927644644715885, 1.673317930679823, 2.185346738127398, 2.835719355561519, 3.202739104159290, 4.227976161858500, 4.531966204186097, 4.964138713872202, 5.335735599914506, 6.296159852389740, 6.817340406785334, 7.235697871095177, 7.850080526128699, 8.338852386295085, 8.480131859412686, 8.978307603905387, 9.664890648726984, 10.31441789601497, 10.74234370632051, 11.05683672654047, 11.57865715952723, 11.99523284199566, 12.59620836225219, 13.22224512103598, 13.57234811854798
