| L(s) = 1 | − 2·3-s − 7-s + 9-s + 2·11-s − 4·13-s + 6·17-s + 4·19-s + 2·21-s + 23-s + 4·27-s − 6·29-s + 4·31-s − 4·33-s − 8·37-s + 8·39-s + 6·41-s − 2·43-s + 8·47-s + 49-s − 12·51-s + 12·53-s − 8·57-s − 6·59-s − 10·61-s − 63-s − 2·67-s − 2·69-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.377·7-s + 1/3·9-s + 0.603·11-s − 1.10·13-s + 1.45·17-s + 0.917·19-s + 0.436·21-s + 0.208·23-s + 0.769·27-s − 1.11·29-s + 0.718·31-s − 0.696·33-s − 1.31·37-s + 1.28·39-s + 0.937·41-s − 0.304·43-s + 1.16·47-s + 1/7·49-s − 1.68·51-s + 1.64·53-s − 1.05·57-s − 0.781·59-s − 1.28·61-s − 0.125·63-s − 0.244·67-s − 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 257600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−12.75499438207900, −12.46394151939387, −12.09230119904622, −11.69597640672803, −11.44408730113124, −10.59477542073552, −10.43595137720477, −9.889953382838631, −9.412610093625409, −9.034092744512258, −8.412826661844517, −7.649174941872209, −7.340168597718675, −6.993545500390645, −6.295543046397027, −5.742065405526937, −5.623314064442541, −4.909598975245652, −4.611921642830310, −3.748667860633498, −3.350744216130007, −2.741038780537191, −2.042130497564654, −1.196662262411567, −0.7556739778173951, 0,
0.7556739778173951, 1.196662262411567, 2.042130497564654, 2.741038780537191, 3.350744216130007, 3.748667860633498, 4.611921642830310, 4.909598975245652, 5.623314064442541, 5.742065405526937, 6.295543046397027, 6.993545500390645, 7.340168597718675, 7.649174941872209, 8.412826661844517, 9.034092744512258, 9.412610093625409, 9.889953382838631, 10.43595137720477, 10.59477542073552, 11.44408730113124, 11.69597640672803, 12.09230119904622, 12.46394151939387, 12.75499438207900
