| L(s) = 1 | − 4·3-s − 12·5-s − 7·7-s − 11·9-s − 11·11-s − 10·13-s + 48·15-s + 24·17-s + 94·19-s + 28·21-s + 180·23-s + 19·25-s + 152·27-s + 30·29-s + 94·31-s + 44·33-s + 84·35-s − 214·37-s + 40·39-s − 48·41-s − 8·43-s + 132·45-s − 30·47-s + 49·49-s − 96·51-s + 54·53-s + 132·55-s + ⋯ |
| L(s) = 1 | − 0.769·3-s − 1.07·5-s − 0.377·7-s − 0.407·9-s − 0.301·11-s − 0.213·13-s + 0.826·15-s + 0.342·17-s + 1.13·19-s + 0.290·21-s + 1.63·23-s + 0.151·25-s + 1.08·27-s + 0.192·29-s + 0.544·31-s + 0.232·33-s + 0.405·35-s − 0.950·37-s + 0.164·39-s − 0.182·41-s − 0.0283·43-s + 0.437·45-s − 0.0931·47-s + 1/7·49-s − 0.263·51-s + 0.139·53-s + 0.323·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1232 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + p T \) |
| 11 | \( 1 + p T \) |
| good | 3 | \( 1 + 4 T + p^{3} T^{2} \) |
| 5 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 10 T + p^{3} T^{2} \) |
| 17 | \( 1 - 24 T + p^{3} T^{2} \) |
| 19 | \( 1 - 94 T + p^{3} T^{2} \) |
| 23 | \( 1 - 180 T + p^{3} T^{2} \) |
| 29 | \( 1 - 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 94 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 + 48 T + p^{3} T^{2} \) |
| 43 | \( 1 + 8 T + p^{3} T^{2} \) |
| 47 | \( 1 + 30 T + p^{3} T^{2} \) |
| 53 | \( 1 - 54 T + p^{3} T^{2} \) |
| 59 | \( 1 + 360 T + p^{3} T^{2} \) |
| 61 | \( 1 + 178 T + p^{3} T^{2} \) |
| 67 | \( 1 - 292 T + p^{3} T^{2} \) |
| 71 | \( 1 + 312 T + p^{3} T^{2} \) |
| 73 | \( 1 - 728 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1288 T + p^{3} T^{2} \) |
| 83 | \( 1 + 66 T + p^{3} T^{2} \) |
| 89 | \( 1 + 390 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1510 T + p^{3} T^{2} \) |
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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
−8.898810348001452132363573358716, −8.025109803198574670524518986577, −7.26362759973626609136523574409, −6.48710014771696171423284068442, −5.40987880033525637221541576690, −4.83484738138951930969294865884, −3.58281124304375225548880065042, −2.83264743735473825515831469058, −0.983273800092516277493884841171, 0,
0.983273800092516277493884841171, 2.83264743735473825515831469058, 3.58281124304375225548880065042, 4.83484738138951930969294865884, 5.40987880033525637221541576690, 6.48710014771696171423284068442, 7.26362759973626609136523574409, 8.025109803198574670524518986577, 8.898810348001452132363573358716
