L(s) = 1 | + 4·2-s + 16·4-s − 233.·7-s + 64·8-s + 89.7·11-s + 209.·13-s − 934.·14-s + 256·16-s + 226.·17-s − 2.18e3·19-s + 358.·22-s + 4.29e3·23-s + 836.·26-s − 3.73e3·28-s − 3.55e3·29-s + 6.90e3·31-s + 1.02e3·32-s + 907.·34-s + 5.43e3·37-s − 8.72e3·38-s − 6.48e3·41-s + 2.19e4·43-s + 1.43e3·44-s + 1.71e4·46-s − 7.71e3·47-s + 3.78e4·49-s + 3.34e3·52-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.80·7-s + 0.353·8-s + 0.223·11-s + 0.343·13-s − 1.27·14-s + 0.250·16-s + 0.190·17-s − 1.38·19-s + 0.158·22-s + 1.69·23-s + 0.242·26-s − 0.901·28-s − 0.785·29-s + 1.28·31-s + 0.176·32-s + 0.134·34-s + 0.652·37-s − 0.979·38-s − 0.602·41-s + 1.81·43-s + 0.111·44-s + 1.19·46-s − 0.509·47-s + 2.24·49-s + 0.171·52-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
\(L(3)\) |
\(\approx\) |
\(2.616321065\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.616321065\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - 4T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + 233.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 89.7T + 1.61e5T^{2} \) |
| 13 | \( 1 - 209.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 226.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 2.18e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.29e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.55e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.90e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 5.43e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.48e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 7.71e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 1.27e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 4.45e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.15e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.96e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 4.64e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.15e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.78e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.09e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 5.13e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.84e4T + 8.58e9T^{2} \) |
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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
−10.34380095262028801744252788382, −9.472392294400421965333790979676, −8.569129416231553700286304682683, −7.13136615487058760042652107265, −6.48466466464573960787323124159, −5.69826068162687972060594813333, −4.33053971757230777690475328511, −3.39702630218179411344179476909, −2.48460259985871881520364235060, −0.73490576426620355996376188422,
0.73490576426620355996376188422, 2.48460259985871881520364235060, 3.39702630218179411344179476909, 4.33053971757230777690475328511, 5.69826068162687972060594813333, 6.48466466464573960787323124159, 7.13136615487058760042652107265, 8.569129416231553700286304682683, 9.472392294400421965333790979676, 10.34380095262028801744252788382