| L(s) = 1 | + 2-s + 4-s + 3·5-s − 7-s + 8-s + 3·10-s − 3·11-s + 2·13-s − 14-s + 16-s − 6·17-s + 2·19-s + 3·20-s − 3·22-s + 4·25-s + 2·26-s − 28-s − 3·29-s − 4·31-s + 32-s − 6·34-s − 3·35-s − 4·37-s + 2·38-s + 3·40-s + 2·43-s − 3·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.34·5-s − 0.377·7-s + 0.353·8-s + 0.948·10-s − 0.904·11-s + 0.554·13-s − 0.267·14-s + 1/4·16-s − 1.45·17-s + 0.458·19-s + 0.670·20-s − 0.639·22-s + 4/5·25-s + 0.392·26-s − 0.188·28-s − 0.557·29-s − 0.718·31-s + 0.176·32-s − 1.02·34-s − 0.507·35-s − 0.657·37-s + 0.324·38-s + 0.474·40-s + 0.304·43-s − 0.452·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 28566 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28566 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−15.43121914619862, −14.84573038048321, −14.33048613427211, −13.62194513919605, −13.41471577909764, −12.96649114427328, −12.60608759788486, −11.65929124243777, −11.19602499182908, −10.63397739094056, −10.15068521019499, −9.526590198511063, −9.079159335167628, −8.351436643980327, −7.720117834133449, −6.754840629170525, −6.635470238653889, −5.866976816513284, −5.284671180934854, −4.987736169433248, −3.972753851694614, −3.414547657756574, −2.488783685917297, −2.156259214189055, −1.319163387741883, 0,
1.319163387741883, 2.156259214189055, 2.488783685917297, 3.414547657756574, 3.972753851694614, 4.987736169433248, 5.284671180934854, 5.866976816513284, 6.635470238653889, 6.754840629170525, 7.720117834133449, 8.351436643980327, 9.079159335167628, 9.526590198511063, 10.15068521019499, 10.63397739094056, 11.19602499182908, 11.65929124243777, 12.60608759788486, 12.96649114427328, 13.41471577909764, 13.62194513919605, 14.33048613427211, 14.84573038048321, 15.43121914619862
