L(s) = 1 | − 7-s − 3·9-s − 6·11-s − 6·13-s − 7·17-s − 2·19-s − 23-s − 5·29-s − 31-s + 5·37-s − 7·41-s + 8·43-s + 8·47-s − 6·49-s − 3·53-s − 13·59-s − 8·61-s + 3·63-s − 9·67-s − 7·71-s + 2·73-s + 6·77-s + 12·79-s + 9·81-s − 5·83-s − 12·89-s + 6·91-s + ⋯ |
L(s) = 1 | − 0.377·7-s − 9-s − 1.80·11-s − 1.66·13-s − 1.69·17-s − 0.458·19-s − 0.208·23-s − 0.928·29-s − 0.179·31-s + 0.821·37-s − 1.09·41-s + 1.21·43-s + 1.16·47-s − 6/7·49-s − 0.412·53-s − 1.69·59-s − 1.02·61-s + 0.377·63-s − 1.09·67-s − 0.830·71-s + 0.234·73-s + 0.683·77-s + 1.35·79-s + 81-s − 0.548·83-s − 1.27·89-s + 0.628·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 + T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 + 13 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−7.22083587290935139526244487725, −6.23410796379957815991608491063, −5.70851045343843158895383847781, −4.87495011254282415463215014177, −4.46941625639081672791082022785, −3.20916294131117857440746408495, −2.54564503274536748546113258107, −2.11057353690086771850592741377, 0, 0,
2.11057353690086771850592741377, 2.54564503274536748546113258107, 3.20916294131117857440746408495, 4.46941625639081672791082022785, 4.87495011254282415463215014177, 5.70851045343843158895383847781, 6.23410796379957815991608491063, 7.22083587290935139526244487725