| L(s) = 1 | + 3-s − 5-s + 2·7-s + 9-s + 2·11-s − 4·13-s − 15-s + 2·17-s + 19-s + 2·21-s + 4·23-s + 25-s + 27-s + 4·29-s + 2·33-s − 2·35-s − 4·39-s + 10·43-s − 45-s − 12·47-s − 3·49-s + 2·51-s − 2·53-s − 2·55-s + 57-s − 4·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.447·5-s + 0.755·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s − 0.258·15-s + 0.485·17-s + 0.229·19-s + 0.436·21-s + 0.834·23-s + 1/5·25-s + 0.192·27-s + 0.742·29-s + 0.348·33-s − 0.338·35-s − 0.640·39-s + 1.52·43-s − 0.149·45-s − 1.75·47-s − 3/7·49-s + 0.280·51-s − 0.274·53-s − 0.269·55-s + 0.132·57-s − 0.520·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.547545683\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.547545683\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 10 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 16 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 16 T + p T^{2} \) |
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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
−8.220833996772994067223729641760, −7.69451426376713427108997802036, −7.07341113368588561296966396700, −6.27483396251328001307537385761, −5.07872552787716309543367621064, −4.69308900565895276645745276376, −3.72176395363745906322596037322, −2.93342543120652714539614603200, −1.97779909070487265858566219387, −0.895818257201027131012503394679,
0.895818257201027131012503394679, 1.97779909070487265858566219387, 2.93342543120652714539614603200, 3.72176395363745906322596037322, 4.69308900565895276645745276376, 5.07872552787716309543367621064, 6.27483396251328001307537385761, 7.07341113368588561296966396700, 7.69451426376713427108997802036, 8.220833996772994067223729641760
