| L(s) = 1 | − 3-s + 5-s − 4·7-s + 9-s − 2·11-s + 2·13-s − 15-s + 4·21-s + 6·23-s + 25-s − 27-s − 31-s + 2·33-s − 4·35-s − 2·37-s − 2·39-s − 10·41-s + 4·43-s + 45-s − 4·47-s + 9·49-s + 6·53-s − 2·55-s + 4·59-s − 4·63-s + 2·65-s − 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.554·13-s − 0.258·15-s + 0.872·21-s + 1.25·23-s + 1/5·25-s − 0.192·27-s − 0.179·31-s + 0.348·33-s − 0.676·35-s − 0.328·37-s − 0.320·39-s − 1.56·41-s + 0.609·43-s + 0.149·45-s − 0.583·47-s + 9/7·49-s + 0.824·53-s − 0.269·55-s + 0.520·59-s − 0.503·63-s + 0.248·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.29105298989983275992068007032, −6.72894435186364961332071079248, −6.23423929862337622286090654579, −5.47802005449241644225921660983, −4.93759904683066851434658135737, −3.80408279527960267052491713845, −3.18634418891858249138409829204, −2.33161002708194670003999099647, −1.10097355699912521743745426647, 0,
1.10097355699912521743745426647, 2.33161002708194670003999099647, 3.18634418891858249138409829204, 3.80408279527960267052491713845, 4.93759904683066851434658135737, 5.47802005449241644225921660983, 6.23423929862337622286090654579, 6.72894435186364961332071079248, 7.29105298989983275992068007032
