| L(s) = 1 | + 2·2-s + 4·4-s + 6·5-s + 8·8-s + 12·10-s − 12·11-s − 38·13-s + 16·16-s − 126·17-s − 20·19-s + 24·20-s − 24·22-s − 168·23-s − 89·25-s − 76·26-s − 30·29-s + 88·31-s + 32·32-s − 252·34-s + 254·37-s − 40·38-s + 48·40-s + 42·41-s − 52·43-s − 48·44-s − 336·46-s − 96·47-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.536·5-s + 0.353·8-s + 0.379·10-s − 0.328·11-s − 0.810·13-s + 1/4·16-s − 1.79·17-s − 0.241·19-s + 0.268·20-s − 0.232·22-s − 1.52·23-s − 0.711·25-s − 0.573·26-s − 0.192·29-s + 0.509·31-s + 0.176·32-s − 1.27·34-s + 1.12·37-s − 0.170·38-s + 0.189·40-s + 0.159·41-s − 0.184·43-s − 0.164·44-s − 1.07·46-s − 0.297·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 12 T + p^{3} T^{2} \) |
| 13 | \( 1 + 38 T + p^{3} T^{2} \) |
| 17 | \( 1 + 126 T + p^{3} T^{2} \) |
| 19 | \( 1 + 20 T + p^{3} T^{2} \) |
| 23 | \( 1 + 168 T + p^{3} T^{2} \) |
| 29 | \( 1 + 30 T + p^{3} T^{2} \) |
| 31 | \( 1 - 88 T + p^{3} T^{2} \) |
| 37 | \( 1 - 254 T + p^{3} T^{2} \) |
| 41 | \( 1 - 42 T + p^{3} T^{2} \) |
| 43 | \( 1 + 52 T + p^{3} T^{2} \) |
| 47 | \( 1 + 96 T + p^{3} T^{2} \) |
| 53 | \( 1 + 198 T + p^{3} T^{2} \) |
| 59 | \( 1 + 660 T + p^{3} T^{2} \) |
| 61 | \( 1 - 538 T + p^{3} T^{2} \) |
| 67 | \( 1 - 884 T + p^{3} T^{2} \) |
| 71 | \( 1 + 792 T + p^{3} T^{2} \) |
| 73 | \( 1 + 218 T + p^{3} T^{2} \) |
| 79 | \( 1 + 520 T + p^{3} T^{2} \) |
| 83 | \( 1 + 492 T + p^{3} T^{2} \) |
| 89 | \( 1 - 810 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1154 T + p^{3} T^{2} \) |
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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
−9.518911918660082050571398852152, −8.403375449194127263142787878014, −7.51388970915789214010217890654, −6.50515740292903229546300246967, −5.86576289761711122784244184709, −4.78469822469035189184723460215, −4.06154560054739478259600664041, −2.62456989302146730330953354145, −1.91771437784758389708284254206, 0,
1.91771437784758389708284254206, 2.62456989302146730330953354145, 4.06154560054739478259600664041, 4.78469822469035189184723460215, 5.86576289761711122784244184709, 6.50515740292903229546300246967, 7.51388970915789214010217890654, 8.403375449194127263142787878014, 9.518911918660082050571398852152
