| L(s) = 1 | − 3-s − 5-s − 4·7-s + 9-s + 4·11-s + 2·13-s + 15-s + 2·17-s − 4·19-s + 4·21-s − 4·23-s + 25-s − 27-s − 2·29-s + 31-s − 4·33-s + 4·35-s − 6·37-s − 2·39-s + 10·41-s + 4·43-s − 45-s − 8·47-s + 9·49-s − 2·51-s − 6·53-s − 4·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s − 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s + 0.872·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 0.371·29-s + 0.179·31-s − 0.696·33-s + 0.676·35-s − 0.986·37-s − 0.320·39-s + 1.56·41-s + 0.609·43-s − 0.149·45-s − 1.16·47-s + 9/7·49-s − 0.280·51-s − 0.824·53-s − 0.539·55-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 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−7.41253253671468841272855879172, −6.56377216446621883786662492969, −6.35830130414138638276261007325, −5.66461981294792900065935727150, −4.59901994427604490879707628500, −3.75614266727194614912987833710, −3.49121505223475884707639091174, −2.23819764675000798726070891958, −1.04527862198316766382488636473, 0,
1.04527862198316766382488636473, 2.23819764675000798726070891958, 3.49121505223475884707639091174, 3.75614266727194614912987833710, 4.59901994427604490879707628500, 5.66461981294792900065935727150, 6.35830130414138638276261007325, 6.56377216446621883786662492969, 7.41253253671468841272855879172
