| L(s) = 1 | + 3-s − 5-s + 9-s − 4·11-s − 2·13-s − 15-s + 6·17-s + 8·23-s + 25-s + 27-s − 10·29-s − 8·31-s − 4·33-s − 2·37-s − 2·39-s + 2·41-s + 8·43-s − 45-s + 4·47-s + 6·51-s − 10·53-s + 4·55-s − 4·59-s − 6·61-s + 2·65-s + 8·69-s + 12·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 1/3·9-s − 1.20·11-s − 0.554·13-s − 0.258·15-s + 1.45·17-s + 1.66·23-s + 1/5·25-s + 0.192·27-s − 1.85·29-s − 1.43·31-s − 0.696·33-s − 0.328·37-s − 0.320·39-s + 0.312·41-s + 1.21·43-s − 0.149·45-s + 0.583·47-s + 0.840·51-s − 1.37·53-s + 0.539·55-s − 0.520·59-s − 0.768·61-s + 0.248·65-s + 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.916050378\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.916050378\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.65387604365781, −14.14023425140060, −13.61673685633762, −12.86504521608144, −12.63708595877652, −12.29610012200487, −11.33316118438534, −10.89611005771600, −10.60473655108564, −9.729584552075993, −9.351748489401567, −8.940881940837945, −8.004581632129311, −7.765344324026310, −7.337328037366579, −6.815435809340372, −5.751063896718677, −5.371762086367650, −4.870097383348957, −4.029094074986213, −3.400898575995148, −2.951557735286377, −2.242746378072034, −1.439312615784487, −0.4747689337625153,
0.4747689337625153, 1.439312615784487, 2.242746378072034, 2.951557735286377, 3.400898575995148, 4.029094074986213, 4.870097383348957, 5.371762086367650, 5.751063896718677, 6.815435809340372, 7.337328037366579, 7.765344324026310, 8.004581632129311, 8.940881940837945, 9.351748489401567, 9.729584552075993, 10.60473655108564, 10.89611005771600, 11.33316118438534, 12.29610012200487, 12.63708595877652, 12.86504521608144, 13.61673685633762, 14.14023425140060, 14.65387604365781
