| L(s) = 1 | + 3·3-s + 8·7-s + 9·9-s − 20·11-s − 22·13-s + 14·17-s − 76·19-s + 24·21-s + 56·23-s + 27·27-s − 154·29-s − 160·31-s − 60·33-s + 162·37-s − 66·39-s − 390·41-s + 388·43-s − 544·47-s − 279·49-s + 42·51-s + 210·53-s − 228·57-s + 380·59-s − 794·61-s + 72·63-s − 148·67-s + 168·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.431·7-s + 1/3·9-s − 0.548·11-s − 0.469·13-s + 0.199·17-s − 0.917·19-s + 0.249·21-s + 0.507·23-s + 0.192·27-s − 0.986·29-s − 0.926·31-s − 0.316·33-s + 0.719·37-s − 0.270·39-s − 1.48·41-s + 1.37·43-s − 1.68·47-s − 0.813·49-s + 0.115·51-s + 0.544·53-s − 0.529·57-s + 0.838·59-s − 1.66·61-s + 0.143·63-s − 0.269·67-s + 0.293·69-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 | \( 1 - 8 T + p^{3} T^{2} \) |
| 11 | \( 1 + 20 T + p^{3} T^{2} \) |
| 13 | \( 1 + 22 T + p^{3} T^{2} \) |
| 17 | \( 1 - 14 T + p^{3} T^{2} \) |
| 19 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 23 | \( 1 - 56 T + p^{3} T^{2} \) |
| 29 | \( 1 + 154 T + p^{3} T^{2} \) |
| 31 | \( 1 + 160 T + p^{3} T^{2} \) |
| 37 | \( 1 - 162 T + p^{3} T^{2} \) |
| 41 | \( 1 + 390 T + p^{3} T^{2} \) |
| 43 | \( 1 - 388 T + p^{3} T^{2} \) |
| 47 | \( 1 + 544 T + p^{3} T^{2} \) |
| 53 | \( 1 - 210 T + p^{3} T^{2} \) |
| 59 | \( 1 - 380 T + p^{3} T^{2} \) |
| 61 | \( 1 + 794 T + p^{3} T^{2} \) |
| 67 | \( 1 + 148 T + p^{3} T^{2} \) |
| 71 | \( 1 - 840 T + p^{3} T^{2} \) |
| 73 | \( 1 + 858 T + p^{3} T^{2} \) |
| 79 | \( 1 + 144 T + p^{3} T^{2} \) |
| 83 | \( 1 - 316 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1098 T + p^{3} T^{2} \) |
| 97 | \( 1 + 994 T + p^{3} T^{2} \) |
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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
−8.955262673280553317262802680710, −8.096307935473738687863221930768, −7.50533795057497973342606760844, −6.57598879362018114083082612481, −5.45060517275198179130568115377, −4.63804477761698937714667170876, −3.60836782431752063195005142257, −2.55165474583805646059960388757, −1.59488653425365061308660157860, 0,
1.59488653425365061308660157860, 2.55165474583805646059960388757, 3.60836782431752063195005142257, 4.63804477761698937714667170876, 5.45060517275198179130568115377, 6.57598879362018114083082612481, 7.50533795057497973342606760844, 8.096307935473738687863221930768, 8.955262673280553317262802680710
