| L(s) = 1 | + 3·3-s + 13·7-s + 9·9-s − 6·11-s + 5·13-s − 78·17-s − 65·19-s + 39·21-s − 138·23-s + 27·27-s + 66·29-s − 299·31-s − 18·33-s − 214·37-s + 15·39-s + 360·41-s − 203·43-s − 78·47-s − 174·49-s − 234·51-s + 636·53-s − 195·57-s − 786·59-s + 467·61-s + 117·63-s + 217·67-s − 414·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.701·7-s + 1/3·9-s − 0.164·11-s + 0.106·13-s − 1.11·17-s − 0.784·19-s + 0.405·21-s − 1.25·23-s + 0.192·27-s + 0.422·29-s − 1.73·31-s − 0.0949·33-s − 0.950·37-s + 0.0615·39-s + 1.37·41-s − 0.719·43-s − 0.242·47-s − 0.507·49-s − 0.642·51-s + 1.64·53-s − 0.453·57-s − 1.73·59-s + 0.980·61-s + 0.233·63-s + 0.395·67-s − 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 7 | \( 1 - 13 T + p^{3} T^{2} \) |
| 11 | \( 1 + 6 T + p^{3} T^{2} \) |
| 13 | \( 1 - 5 T + p^{3} T^{2} \) |
| 17 | \( 1 + 78 T + p^{3} T^{2} \) |
| 19 | \( 1 + 65 T + p^{3} T^{2} \) |
| 23 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 29 | \( 1 - 66 T + p^{3} T^{2} \) |
| 31 | \( 1 + 299 T + p^{3} T^{2} \) |
| 37 | \( 1 + 214 T + p^{3} T^{2} \) |
| 41 | \( 1 - 360 T + p^{3} T^{2} \) |
| 43 | \( 1 + 203 T + p^{3} T^{2} \) |
| 47 | \( 1 + 78 T + p^{3} T^{2} \) |
| 53 | \( 1 - 12 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 786 T + p^{3} T^{2} \) |
| 61 | \( 1 - 467 T + p^{3} T^{2} \) |
| 67 | \( 1 - 217 T + p^{3} T^{2} \) |
| 71 | \( 1 - 360 T + p^{3} T^{2} \) |
| 73 | \( 1 + 286 T + p^{3} T^{2} \) |
| 79 | \( 1 + 272 T + p^{3} T^{2} \) |
| 83 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 89 | \( 1 + p^{3} T^{2} \) |
| 97 | \( 1 + 511 T + p^{3} T^{2} \) |
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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
−8.803500134319591274932351104780, −8.278728828930639424910110952894, −7.42691537302626434831512982401, −6.56105726080252807587063328287, −5.52739754439847007169892125460, −4.49560752754506680873337117645, −3.75979948387020228305105931149, −2.42930041771837526207141840307, −1.66105556554310891302434905547, 0,
1.66105556554310891302434905547, 2.42930041771837526207141840307, 3.75979948387020228305105931149, 4.49560752754506680873337117645, 5.52739754439847007169892125460, 6.56105726080252807587063328287, 7.42691537302626434831512982401, 8.278728828930639424910110952894, 8.803500134319591274932351104780
