L(s) = 1 | + 3-s − 2·5-s + 9-s + 11-s − 2·13-s − 2·15-s + 17-s + 4·19-s − 25-s + 27-s − 2·29-s + 8·31-s + 33-s − 10·37-s − 2·39-s − 6·41-s − 4·43-s − 2·45-s − 8·47-s − 7·49-s + 51-s + 6·53-s − 2·55-s + 4·57-s + 12·59-s − 10·61-s + 4·65-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1/3·9-s + 0.301·11-s − 0.554·13-s − 0.516·15-s + 0.242·17-s + 0.917·19-s − 1/5·25-s + 0.192·27-s − 0.371·29-s + 1.43·31-s + 0.174·33-s − 1.64·37-s − 0.320·39-s − 0.937·41-s − 0.609·43-s − 0.298·45-s − 1.16·47-s − 49-s + 0.140·51-s + 0.824·53-s − 0.269·55-s + 0.529·57-s + 1.56·59-s − 1.28·61-s + 0.496·65-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 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−7.49000926382861827247562403171, −6.91011895808126925547297073125, −6.14475954207001671385750878889, −5.08103683632015531320429372736, −4.64368734654277400104943952108, −3.55410006291023208410029133033, −3.34696242752996878598902878796, −2.24737947675728240789262397920, −1.27236489460320032405296651590, 0,
1.27236489460320032405296651590, 2.24737947675728240789262397920, 3.34696242752996878598902878796, 3.55410006291023208410029133033, 4.64368734654277400104943952108, 5.08103683632015531320429372736, 6.14475954207001671385750878889, 6.91011895808126925547297073125, 7.49000926382861827247562403171