| L(s) = 1 | − 2-s + 4-s + 2·5-s − 8-s − 2·10-s + 3·11-s + 2·13-s + 16-s − 17-s − 8·19-s + 2·20-s − 3·22-s − 23-s − 25-s − 2·26-s + 29-s − 2·31-s − 32-s + 34-s + 10·37-s + 8·38-s − 2·40-s − 10·41-s + 6·43-s + 3·44-s + 46-s + 3·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s − 0.353·8-s − 0.632·10-s + 0.904·11-s + 0.554·13-s + 1/4·16-s − 0.242·17-s − 1.83·19-s + 0.447·20-s − 0.639·22-s − 0.208·23-s − 1/5·25-s − 0.392·26-s + 0.185·29-s − 0.359·31-s − 0.176·32-s + 0.171·34-s + 1.64·37-s + 1.29·38-s − 0.316·40-s − 1.56·41-s + 0.914·43-s + 0.452·44-s + 0.147·46-s + 0.437·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 13 T + p T^{2} \) |
| 79 | \( 1 - 15 T + p T^{2} \) |
| 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
−16.04239796923134, −15.39222020412985, −14.79547900988473, −14.43485458367706, −13.64039251400437, −13.30105941289244, −12.54861571030463, −12.12710162766386, −11.20477848979415, −11.01984415702867, −10.24397538739891, −9.821241741693774, −9.078616669513007, −8.864682542550758, −8.124138172161338, −7.514378748625588, −6.634777561251201, −6.194254891330721, −5.953166899226322, −4.841009177289289, −4.188681329346450, −3.437468572339283, −2.457238235943097, −1.873129551214916, −1.197038885161425, 0,
1.197038885161425, 1.873129551214916, 2.457238235943097, 3.437468572339283, 4.188681329346450, 4.841009177289289, 5.953166899226322, 6.194254891330721, 6.634777561251201, 7.514378748625588, 8.124138172161338, 8.864682542550758, 9.078616669513007, 9.821241741693774, 10.24397538739891, 11.01984415702867, 11.20477848979415, 12.12710162766386, 12.54861571030463, 13.30105941289244, 13.64039251400437, 14.43485458367706, 14.79547900988473, 15.39222020412985, 16.04239796923134
