L(s) = 1 | − 3-s − 2·7-s + 9-s − 11-s − 4·13-s − 17-s − 8·19-s + 2·21-s − 6·23-s − 5·25-s − 27-s + 6·29-s + 4·31-s + 33-s + 2·37-s + 4·39-s − 6·41-s + 4·43-s − 6·47-s − 3·49-s + 51-s − 12·53-s + 8·57-s − 4·61-s − 2·63-s + 4·67-s + 6·69-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.755·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.242·17-s − 1.83·19-s + 0.436·21-s − 1.25·23-s − 25-s − 0.192·27-s + 1.11·29-s + 0.718·31-s + 0.174·33-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s − 0.875·47-s − 3/7·49-s + 0.140·51-s − 1.64·53-s + 1.05·57-s − 0.512·61-s − 0.251·63-s + 0.488·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8976 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8976 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.4054843869\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4054843869\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
good | 5 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−7.84101836897999624714812591524, −6.81954299628327760911748405030, −6.36853399273626767544726680898, −5.88416962585052911008444132407, −4.80833183429404885729193904251, −4.44445241989965611558348767101, −3.49941593833619035184009093732, −2.53707363053865273434426886769, −1.83983556355970219053051070112, −0.29814890818923373642323340894,
0.29814890818923373642323340894, 1.83983556355970219053051070112, 2.53707363053865273434426886769, 3.49941593833619035184009093732, 4.44445241989965611558348767101, 4.80833183429404885729193904251, 5.88416962585052911008444132407, 6.36853399273626767544726680898, 6.81954299628327760911748405030, 7.84101836897999624714812591524