| L(s) = 1 | − 2·4-s + 3·5-s − 4·7-s − 11-s + 2·13-s + 4·16-s + 6·17-s + 2·19-s − 6·20-s − 3·23-s + 4·25-s + 8·28-s + 6·29-s + 8·31-s − 12·35-s + 2·37-s + 8·43-s + 2·44-s − 3·47-s + 9·49-s − 4·52-s − 3·53-s − 3·55-s + 8·61-s − 8·64-s + 6·65-s − 13·67-s + ⋯ |
| L(s) = 1 | − 4-s + 1.34·5-s − 1.51·7-s − 0.301·11-s + 0.554·13-s + 16-s + 1.45·17-s + 0.458·19-s − 1.34·20-s − 0.625·23-s + 4/5·25-s + 1.51·28-s + 1.11·29-s + 1.43·31-s − 2.02·35-s + 0.328·37-s + 1.21·43-s + 0.301·44-s − 0.437·47-s + 9/7·49-s − 0.554·52-s − 0.412·53-s − 0.404·55-s + 1.02·61-s − 64-s + 0.744·65-s − 1.58·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 891 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.382930520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.382930520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 2 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 3 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 - 18 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 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
−9.944381223757953089753372944188, −9.543697124721827346113055813271, −8.666250032155508275985322036872, −7.67579797116786415531719850377, −6.28320198661316535576204964999, −5.93548013183253829968096634124, −4.94627004773663169249134422915, −3.64232856897985226857412108530, −2.75315253823480622918526608173, −0.985102124491110264282669993364,
0.985102124491110264282669993364, 2.75315253823480622918526608173, 3.64232856897985226857412108530, 4.94627004773663169249134422915, 5.93548013183253829968096634124, 6.28320198661316535576204964999, 7.67579797116786415531719850377, 8.666250032155508275985322036872, 9.543697124721827346113055813271, 9.944381223757953089753372944188
