| L(s) = 1 | − 3·3-s + 7·7-s + 9·9-s + 54·11-s − 14·13-s + 44·17-s − 98·19-s − 21·21-s − 8·23-s − 27·27-s − 146·29-s − 254·31-s − 162·33-s − 68·37-s + 42·39-s − 270·41-s − 172·43-s + 328·47-s + 49·49-s − 132·51-s + 594·53-s + 294·57-s + 376·59-s + 318·61-s + 63·63-s + 296·67-s + 24·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.377·7-s + 1/3·9-s + 1.48·11-s − 0.298·13-s + 0.627·17-s − 1.18·19-s − 0.218·21-s − 0.0725·23-s − 0.192·27-s − 0.934·29-s − 1.47·31-s − 0.854·33-s − 0.302·37-s + 0.172·39-s − 1.02·41-s − 0.609·43-s + 1.01·47-s + 1/7·49-s − 0.362·51-s + 1.53·53-s + 0.683·57-s + 0.829·59-s + 0.667·61-s + 0.125·63-s + 0.539·67-s + 0.0418·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2100 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 11 | \( 1 - 54 T + p^{3} T^{2} \) |
| 13 | \( 1 + 14 T + p^{3} T^{2} \) |
| 17 | \( 1 - 44 T + p^{3} T^{2} \) |
| 19 | \( 1 + 98 T + p^{3} T^{2} \) |
| 23 | \( 1 + 8 T + p^{3} T^{2} \) |
| 29 | \( 1 + 146 T + p^{3} T^{2} \) |
| 31 | \( 1 + 254 T + p^{3} T^{2} \) |
| 37 | \( 1 + 68 T + p^{3} T^{2} \) |
| 41 | \( 1 + 270 T + p^{3} T^{2} \) |
| 43 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 47 | \( 1 - 328 T + p^{3} T^{2} \) |
| 53 | \( 1 - 594 T + p^{3} T^{2} \) |
| 59 | \( 1 - 376 T + p^{3} T^{2} \) |
| 61 | \( 1 - 318 T + p^{3} T^{2} \) |
| 67 | \( 1 - 296 T + p^{3} T^{2} \) |
| 71 | \( 1 + 378 T + p^{3} T^{2} \) |
| 73 | \( 1 + 1138 T + p^{3} T^{2} \) |
| 79 | \( 1 - 284 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1032 T + p^{3} T^{2} \) |
| 89 | \( 1 - 454 T + p^{3} T^{2} \) |
| 97 | \( 1 + 678 T + p^{3} T^{2} \) |
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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
−8.510137287715282988558499739402, −7.36791088613519186346042615364, −6.85186479557852006489295972483, −5.93303925816290057304935156144, −5.26453642083116929215971219879, −4.20853222681836458196141188402, −3.63110735403181355364510587998, −2.11492855807104466119272253112, −1.26119411876732103112710977033, 0,
1.26119411876732103112710977033, 2.11492855807104466119272253112, 3.63110735403181355364510587998, 4.20853222681836458196141188402, 5.26453642083116929215971219879, 5.93303925816290057304935156144, 6.85186479557852006489295972483, 7.36791088613519186346042615364, 8.510137287715282988558499739402
