| L(s) = 1 | + 3-s + 5-s + 4·7-s + 9-s − 4·11-s + 2·13-s + 15-s − 6·17-s + 19-s + 4·21-s + 4·23-s + 25-s + 27-s + 6·29-s + 8·31-s − 4·33-s + 4·35-s + 10·37-s + 2·39-s + 2·41-s + 45-s − 4·47-s + 9·49-s − 6·51-s + 2·53-s − 4·55-s + 57-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 0.258·15-s − 1.45·17-s + 0.229·19-s + 0.872·21-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 + 0.676·35-s + 1.64·37-s + 0.320·39-s + 0.312·41-s + 0.149·45-s − 0.583·47-s + 9/7·49-s − 0.840·51-s + 0.274·53-s − 0.539·55-s + 0.132·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.837185025\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.837185025\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−8.821553412971676984605893479951, −8.256991610211039245481488301316, −7.73265002842748370944954044002, −6.75396098282621610633138578491, −5.84626419669606334420033762910, −4.74657644198197122956787738437, −4.51106524290148230742370733061, −2.93815846477151288410438007131, −2.26230374822591998955448290486, −1.15646988225009570825242814423,
1.15646988225009570825242814423, 2.26230374822591998955448290486, 2.93815846477151288410438007131, 4.51106524290148230742370733061, 4.74657644198197122956787738437, 5.84626419669606334420033762910, 6.75396098282621610633138578491, 7.73265002842748370944954044002, 8.256991610211039245481488301316, 8.821553412971676984605893479951
