| L(s) = 1 | − 3-s + 5-s + 2·7-s + 9-s − 4·11-s − 15-s + 17-s − 4·19-s − 2·21-s − 4·23-s + 25-s − 27-s + 6·29-s + 8·31-s + 4·33-s + 2·35-s − 6·37-s + 8·41-s − 2·43-s + 45-s + 8·47-s − 3·49-s − 51-s + 14·53-s − 4·55-s + 4·57-s − 6·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.755·7-s + 1/3·9-s − 1.20·11-s − 0.258·15-s + 0.242·17-s − 0.917·19-s − 0.436·21-s − 0.834·23-s + 1/5·25-s − 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s + 0.338·35-s − 0.986·37-s + 1.24·41-s − 0.304·43-s + 0.149·45-s + 1.16·47-s − 3/7·49-s − 0.140·51-s + 1.92·53-s − 0.539·55-s + 0.529·57-s − 0.781·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.624919003\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.624919003\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.244058657971264727455111667021, −7.85881881801700894491882667041, −6.86833485055146344043791758545, −6.15261556486240595427602564877, −5.42613862498967272760721056575, −4.80788694117905551244666391229, −4.06125348432763866604383893942, −2.73224729177077021226526060036, −1.99398978518827439771652660601, −0.74713672056689960984886962980,
0.74713672056689960984886962980, 1.99398978518827439771652660601, 2.73224729177077021226526060036, 4.06125348432763866604383893942, 4.80788694117905551244666391229, 5.42613862498967272760721056575, 6.15261556486240595427602564877, 6.86833485055146344043791758545, 7.85881881801700894491882667041, 8.244058657971264727455111667021
