| L(s) = 1 | − 3-s + 5-s + 7-s − 2·9-s − 4·11-s − 15-s − 6·17-s − 4·19-s − 21-s − 23-s − 4·25-s + 5·27-s + 6·29-s + 6·31-s + 4·33-s + 35-s − 4·37-s + 8·41-s − 4·43-s − 2·45-s + 6·47-s + 49-s + 6·51-s − 4·53-s − 4·55-s + 4·57-s − 3·59-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 0.377·7-s − 2/3·9-s − 1.20·11-s − 0.258·15-s − 1.45·17-s − 0.917·19-s − 0.218·21-s − 0.208·23-s − 4/5·25-s + 0.962·27-s + 1.11·29-s + 1.07·31-s + 0.696·33-s + 0.169·35-s − 0.657·37-s + 1.24·41-s − 0.609·43-s − 0.298·45-s + 0.875·47-s + 1/7·49-s + 0.840·51-s − 0.549·53-s − 0.539·55-s + 0.529·57-s − 0.390·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 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
−14.11495825806695, −13.78895598978217, −13.41399151210823, −12.74795306224321, −12.31497613763570, −11.82144979146461, −11.12173831215163, −10.89166869808562, −10.42859937932330, −9.905256459797968, −9.212808816399962, −8.689900532514724, −8.114994829138697, −7.898440893596246, −6.887097561686372, −6.521720370160890, −5.973278848859629, −5.497284814242660, −4.827928250916747, −4.530767963393382, −3.746727586010740, −2.717245844386389, −2.468455562384726, −1.789349596543087, −0.7115513838955087, 0,
0.7115513838955087, 1.789349596543087, 2.468455562384726, 2.717245844386389, 3.746727586010740, 4.530767963393382, 4.827928250916747, 5.497284814242660, 5.973278848859629, 6.521720370160890, 6.887097561686372, 7.898440893596246, 8.114994829138697, 8.689900532514724, 9.212808816399962, 9.905256459797968, 10.42859937932330, 10.89166869808562, 11.12173831215163, 11.82144979146461, 12.31497613763570, 12.74795306224321, 13.41399151210823, 13.78895598978217, 14.11495825806695
