| L(s) = 1 | + 5-s − 3·7-s − 11-s + 6·17-s + 8·19-s − 23-s + 25-s − 3·29-s + 31-s − 3·35-s − 10·37-s − 5·41-s − 43-s + 7·47-s + 2·49-s − 5·53-s − 55-s − 8·61-s − 4·67-s − 2·71-s − 9·73-s + 3·77-s + 10·79-s + 6·85-s + 8·89-s + 8·95-s + 18·97-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s − 0.301·11-s + 1.45·17-s + 1.83·19-s − 0.208·23-s + 1/5·25-s − 0.557·29-s + 0.179·31-s − 0.507·35-s − 1.64·37-s − 0.780·41-s − 0.152·43-s + 1.02·47-s + 2/7·49-s − 0.686·53-s − 0.134·55-s − 1.02·61-s − 0.488·67-s − 0.237·71-s − 1.05·73-s + 0.341·77-s + 1.12·79-s + 0.650·85-s + 0.847·89-s + 0.820·95-s + 1.82·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 334620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.831080838\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.831080838\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−12.47008002635249, −12.12560035205131, −11.90223084849779, −11.26751350342197, −10.53604020006924, −10.27067599321017, −9.904245482657793, −9.398764698386016, −9.124540570654466, −8.540253213444665, −7.797362366908105, −7.545223467645339, −7.065090399369821, −6.512595052118484, −5.949890611279986, −5.619004322081363, −5.131607505606399, −4.689047981959461, −3.707332945419413, −3.381494904622858, −3.097071678059826, −2.421964718060379, −1.636596275851372, −1.152729775726102, −0.3705010852934017,
0.3705010852934017, 1.152729775726102, 1.636596275851372, 2.421964718060379, 3.097071678059826, 3.381494904622858, 3.707332945419413, 4.689047981959461, 5.131607505606399, 5.619004322081363, 5.949890611279986, 6.512595052118484, 7.065090399369821, 7.545223467645339, 7.797362366908105, 8.540253213444665, 9.124540570654466, 9.398764698386016, 9.904245482657793, 10.27067599321017, 10.53604020006924, 11.26751350342197, 11.90223084849779, 12.12560035205131, 12.47008002635249
