| L(s) = 1 | + 3-s − 5-s − 3·7-s + 9-s + 3·11-s − 5·13-s − 15-s + 5·17-s − 19-s − 3·21-s + 7·23-s + 25-s + 27-s − 6·29-s − 6·31-s + 3·33-s + 3·35-s − 37-s − 5·39-s + 8·41-s − 4·43-s − 45-s + 12·47-s + 2·49-s + 5·51-s − 13·53-s − 3·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s − 1.13·7-s + 1/3·9-s + 0.904·11-s − 1.38·13-s − 0.258·15-s + 1.21·17-s − 0.229·19-s − 0.654·21-s + 1.45·23-s + 1/5·25-s + 0.192·27-s − 1.11·29-s − 1.07·31-s + 0.522·33-s + 0.507·35-s − 0.164·37-s − 0.800·39-s + 1.24·41-s − 0.609·43-s − 0.149·45-s + 1.75·47-s + 2/7·49-s + 0.700·51-s − 1.78·53-s − 0.404·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + 13 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−7.47152372051095145887790196585, −6.87652874797842303261208991452, −6.15106317018233803395432974930, −5.29055618782402605426652372976, −4.51425707376747966944199919992, −3.60418219822449164516095491680, −3.22715282926210461631757443233, −2.36565669464103782956802817912, −1.22696547943052363437671750768, 0,
1.22696547943052363437671750768, 2.36565669464103782956802817912, 3.22715282926210461631757443233, 3.60418219822449164516095491680, 4.51425707376747966944199919992, 5.29055618782402605426652372976, 6.15106317018233803395432974930, 6.87652874797842303261208991452, 7.47152372051095145887790196585
