| L(s) = 1 | + 2-s + 4-s + 7-s + 8-s + 4·11-s + 13-s + 14-s + 16-s − 17-s + 5·19-s + 4·22-s + 4·23-s + 26-s + 28-s − 3·29-s − 8·31-s + 32-s − 34-s + 4·37-s + 5·38-s + 2·41-s + 4·44-s + 4·46-s + 7·47-s + 49-s + 52-s − 9·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.377·7-s + 0.353·8-s + 1.20·11-s + 0.277·13-s + 0.267·14-s + 1/4·16-s − 0.242·17-s + 1.14·19-s + 0.852·22-s + 0.834·23-s + 0.196·26-s + 0.188·28-s − 0.557·29-s − 1.43·31-s + 0.176·32-s − 0.171·34-s + 0.657·37-s + 0.811·38-s + 0.312·41-s + 0.603·44-s + 0.589·46-s + 1.02·47-s + 1/7·49-s + 0.138·52-s − 1.23·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.290586939\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.290586939\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| good | 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 + 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−7.44841161262770126290287348937, −7.02697719448467560715684714467, −6.26950333267827545061043977776, −5.54829818680904885660066714822, −5.00818981735505829019848102734, −4.06902507210204378860473736464, −3.65188227260212594501278346629, −2.73485277110470558894976584456, −1.75485318531615352633902535524, −0.952901588319982110631446613793,
0.952901588319982110631446613793, 1.75485318531615352633902535524, 2.73485277110470558894976584456, 3.65188227260212594501278346629, 4.06902507210204378860473736464, 5.00818981735505829019848102734, 5.54829818680904885660066714822, 6.26950333267827545061043977776, 7.02697719448467560715684714467, 7.44841161262770126290287348937
