| L(s) = 1 | + 4·2-s + 16·4-s + 233·7-s + 64·8-s + 498·11-s + 809·13-s + 932·14-s + 256·16-s + 1.00e3·17-s − 1.70e3·19-s + 1.99e3·22-s − 1.55e3·23-s + 3.23e3·26-s + 3.72e3·28-s − 7.83e3·29-s + 977·31-s + 1.02e3·32-s + 4.00e3·34-s − 4.82e3·37-s − 6.82e3·38-s + 8.14e3·41-s + 1.94e4·43-s + 7.96e3·44-s − 6.21e3·46-s − 8.41e3·47-s + 3.74e4·49-s + 1.29e4·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.79·7-s + 0.353·8-s + 1.24·11-s + 1.32·13-s + 1.27·14-s + 1/4·16-s + 0.840·17-s − 1.08·19-s + 0.877·22-s − 0.612·23-s + 0.938·26-s + 0.898·28-s − 1.72·29-s + 0.182·31-s + 0.176·32-s + 0.594·34-s − 0.579·37-s − 0.766·38-s + 0.756·41-s + 1.60·43-s + 0.620·44-s − 0.433·46-s − 0.555·47-s + 2.23·49-s + 0.663·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(5.104672805\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.104672805\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p^{2} T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 233 T + p^{5} T^{2} \) |
| 11 | \( 1 - 498 T + p^{5} T^{2} \) |
| 13 | \( 1 - 809 T + p^{5} T^{2} \) |
| 17 | \( 1 - 1002 T + p^{5} T^{2} \) |
| 19 | \( 1 + 1705 T + p^{5} T^{2} \) |
| 23 | \( 1 + 1554 T + p^{5} T^{2} \) |
| 29 | \( 1 + 270 p T + p^{5} T^{2} \) |
| 31 | \( 1 - 977 T + p^{5} T^{2} \) |
| 37 | \( 1 + 4822 T + p^{5} T^{2} \) |
| 41 | \( 1 - 8148 T + p^{5} T^{2} \) |
| 43 | \( 1 - 19469 T + p^{5} T^{2} \) |
| 47 | \( 1 + 8418 T + p^{5} T^{2} \) |
| 53 | \( 1 + 17664 T + p^{5} T^{2} \) |
| 59 | \( 1 + 35910 T + p^{5} T^{2} \) |
| 61 | \( 1 - 3527 T + p^{5} T^{2} \) |
| 67 | \( 1 - 57473 T + p^{5} T^{2} \) |
| 71 | \( 1 - 7548 T + p^{5} T^{2} \) |
| 73 | \( 1 + 646 T + p^{5} T^{2} \) |
| 79 | \( 1 + 22720 T + p^{5} T^{2} \) |
| 83 | \( 1 + 11574 T + p^{5} T^{2} \) |
| 89 | \( 1 - 78960 T + p^{5} T^{2} \) |
| 97 | \( 1 - 54593 T + p^{5} T^{2} \) |
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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
−10.75713483642212488604977906849, −9.270227088107908255116491662162, −8.332537307322987143897240047172, −7.58051013252183669005009587067, −6.32551702304577183083081752209, −5.52663494274947253630391341904, −4.35895737022033181119838335935, −3.71154312204968321758236996928, −1.96121770350680028776492961346, −1.20980910060563219141429428916,
1.20980910060563219141429428916, 1.96121770350680028776492961346, 3.71154312204968321758236996928, 4.35895737022033181119838335935, 5.52663494274947253630391341904, 6.32551702304577183083081752209, 7.58051013252183669005009587067, 8.332537307322987143897240047172, 9.270227088107908255116491662162, 10.75713483642212488604977906849
