| L(s) = 1 | − 3·3-s + 5·5-s + 7·7-s + 9·9-s + 44·11-s + 22·13-s − 15·15-s + 66·17-s + 132·19-s − 21·21-s + 168·23-s + 25·25-s − 27·27-s + 54·29-s − 144·31-s − 132·33-s + 35·35-s − 354·37-s − 66·39-s − 22·41-s + 156·43-s + 45·45-s + 240·47-s + 49·49-s − 198·51-s − 354·53-s + 220·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s + 1/3·9-s + 1.20·11-s + 0.469·13-s − 0.258·15-s + 0.941·17-s + 1.59·19-s − 0.218·21-s + 1.52·23-s + 1/5·25-s − 0.192·27-s + 0.345·29-s − 0.834·31-s − 0.696·33-s + 0.169·35-s − 1.57·37-s − 0.270·39-s − 0.0838·41-s + 0.553·43-s + 0.149·45-s + 0.744·47-s + 1/7·49-s − 0.543·51-s − 0.917·53-s + 0.539·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1680 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.774455331\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.774455331\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 - 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 - 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 132 T + p^{3} T^{2} \) |
| 23 | \( 1 - 168 T + p^{3} T^{2} \) |
| 29 | \( 1 - 54 T + p^{3} T^{2} \) |
| 31 | \( 1 + 144 T + p^{3} T^{2} \) |
| 37 | \( 1 + 354 T + p^{3} T^{2} \) |
| 41 | \( 1 + 22 T + p^{3} T^{2} \) |
| 43 | \( 1 - 156 T + p^{3} T^{2} \) |
| 47 | \( 1 - 240 T + p^{3} T^{2} \) |
| 53 | \( 1 + 354 T + p^{3} T^{2} \) |
| 59 | \( 1 - 76 T + p^{3} T^{2} \) |
| 61 | \( 1 + 154 T + p^{3} T^{2} \) |
| 67 | \( 1 - 628 T + p^{3} T^{2} \) |
| 71 | \( 1 + 8 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1018 T + p^{3} T^{2} \) |
| 79 | \( 1 + 96 T + p^{3} T^{2} \) |
| 83 | \( 1 + 348 T + p^{3} T^{2} \) |
| 89 | \( 1 - 218 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1598 T + p^{3} T^{2} \) |
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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
−9.224933170266277920763151822230, −8.209782858475346159822091807084, −7.19059251922942537910693660997, −6.65608105088371335036045915190, −5.55350006728832410311651845114, −5.17097211149423665592866785225, −3.94663510513296687634476716750, −3.08207624238810424877620767743, −1.53725449547197032203000705969, −0.936941124256024696352504300174,
0.936941124256024696352504300174, 1.53725449547197032203000705969, 3.08207624238810424877620767743, 3.94663510513296687634476716750, 5.17097211149423665592866785225, 5.55350006728832410311651845114, 6.65608105088371335036045915190, 7.19059251922942537910693660997, 8.209782858475346159822091807084, 9.224933170266277920763151822230
