L(s) = 1 | − 3-s − 2·5-s + 4·7-s + 9-s + 11-s − 4·13-s + 2·15-s − 17-s + 8·19-s − 4·21-s − 25-s − 27-s − 10·31-s − 33-s − 8·35-s + 8·37-s + 4·39-s − 10·41-s + 8·43-s − 2·45-s − 10·47-s + 9·49-s + 51-s − 12·53-s − 2·55-s − 8·57-s + 8·59-s + ⋯ |
L(s) = 1 | − 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s + 0.301·11-s − 1.10·13-s + 0.516·15-s − 0.242·17-s + 1.83·19-s − 0.872·21-s − 1/5·25-s − 0.192·27-s − 1.79·31-s − 0.174·33-s − 1.35·35-s + 1.31·37-s + 0.640·39-s − 1.56·41-s + 1.21·43-s − 0.298·45-s − 1.45·47-s + 9/7·49-s + 0.140·51-s − 1.64·53-s − 0.269·55-s − 1.05·57-s + 1.04·59-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 4 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−7.46359276400536125773428488630, −6.97583120755444883778376195542, −5.85781015752216054655159126448, −5.15974362930734467759515713178, −4.74931154947459423626138869878, −4.00901430742252610755757356422, −3.16828699561523386551113421727, −2.00575495345830670442783676809, −1.19646942560819514176459923997, 0,
1.19646942560819514176459923997, 2.00575495345830670442783676809, 3.16828699561523386551113421727, 4.00901430742252610755757356422, 4.74931154947459423626138869878, 5.15974362930734467759515713178, 5.85781015752216054655159126448, 6.97583120755444883778376195542, 7.46359276400536125773428488630