| L(s) = 1 | − 3·3-s − 5·5-s + 16·7-s + 9·9-s − 28·11-s + 26·13-s + 15·15-s − 62·17-s − 68·19-s − 48·21-s + 208·23-s + 25·25-s − 27·27-s + 58·29-s − 160·31-s + 84·33-s − 80·35-s − 270·37-s − 78·39-s + 282·41-s + 76·43-s − 45·45-s + 280·47-s − 87·49-s + 186·51-s + 210·53-s + 140·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 0.863·7-s + 1/3·9-s − 0.767·11-s + 0.554·13-s + 0.258·15-s − 0.884·17-s − 0.821·19-s − 0.498·21-s + 1.88·23-s + 1/5·25-s − 0.192·27-s + 0.371·29-s − 0.926·31-s + 0.443·33-s − 0.386·35-s − 1.19·37-s − 0.320·39-s + 1.07·41-s + 0.269·43-s − 0.149·45-s + 0.868·47-s − 0.253·49-s + 0.510·51-s + 0.544·53-s + 0.343·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 960 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + p T \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 - 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 17 | \( 1 + 62 T + p^{3} T^{2} \) |
| 19 | \( 1 + 68 T + p^{3} T^{2} \) |
| 23 | \( 1 - 208 T + p^{3} T^{2} \) |
| 29 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 + 270 T + p^{3} T^{2} \) |
| 41 | \( 1 - 282 T + p^{3} T^{2} \) |
| 43 | \( 1 - 76 T + p^{3} T^{2} \) |
| 47 | \( 1 - 280 T + p^{3} T^{2} \) |
| 53 | \( 1 - 210 T + p^{3} T^{2} \) |
| 59 | \( 1 - 196 T + p^{3} T^{2} \) |
| 61 | \( 1 + 742 T + p^{3} T^{2} \) |
| 67 | \( 1 - 836 T + p^{3} T^{2} \) |
| 71 | \( 1 - 504 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1062 T + p^{3} T^{2} \) |
| 79 | \( 1 + 768 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1052 T + p^{3} T^{2} \) |
| 89 | \( 1 + 726 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1406 T + p^{3} T^{2} \) |
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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
−9.022164909958282862153992746229, −8.460919227824053169598772169415, −7.45702172147835434338903135840, −6.76217086289775134549806564771, −5.61484356089664864121924228784, −4.84914411646464727924759528734, −4.02303799054745753980982452300, −2.64052435080914712108732070476, −1.32674191206636419163941515341, 0,
1.32674191206636419163941515341, 2.64052435080914712108732070476, 4.02303799054745753980982452300, 4.84914411646464727924759528734, 5.61484356089664864121924228784, 6.76217086289775134549806564771, 7.45702172147835434338903135840, 8.460919227824053169598772169415, 9.022164909958282862153992746229
