| L(s) = 1 | + 3-s − 4·7-s − 2·9-s + 4·13-s + 4·17-s − 4·19-s − 4·21-s − 3·23-s − 5·27-s − 8·29-s − 9·31-s + 5·37-s + 4·39-s + 12·41-s − 8·43-s + 4·47-s + 9·49-s + 4·51-s + 10·53-s − 4·57-s − 7·59-s + 8·61-s + 8·63-s + 11·67-s − 3·69-s + 9·71-s + 4·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.51·7-s − 2/3·9-s + 1.10·13-s + 0.970·17-s − 0.917·19-s − 0.872·21-s − 0.625·23-s − 0.962·27-s − 1.48·29-s − 1.61·31-s + 0.821·37-s + 0.640·39-s + 1.87·41-s − 1.21·43-s + 0.583·47-s + 9/7·49-s + 0.560·51-s + 1.37·53-s − 0.529·57-s − 0.911·59-s + 1.02·61-s + 1.00·63-s + 1.34·67-s − 0.361·69-s + 1.06·71-s + 0.468·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 9 T + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 7 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 - 9 T + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−14.76104822497822, −14.34845176036439, −13.73195896739419, −13.27933591380579, −12.79167228195650, −12.50466445136004, −11.73566716865417, −11.02246893136330, −10.84241950475522, −9.953134996323508, −9.531171102985752, −9.143893389167621, −8.571944986625610, −8.024383740948529, −7.465915357459579, −6.814084220661893, −6.127847855291421, −5.816271231153441, −5.313878305883958, −4.063681629357419, −3.691111107958831, −3.359714921697935, −2.492470316820657, −1.996959155722319, −0.8516756205449636, 0,
0.8516756205449636, 1.996959155722319, 2.492470316820657, 3.359714921697935, 3.691111107958831, 4.063681629357419, 5.313878305883958, 5.816271231153441, 6.127847855291421, 6.814084220661893, 7.465915357459579, 8.024383740948529, 8.571944986625610, 9.143893389167621, 9.531171102985752, 9.953134996323508, 10.84241950475522, 11.02246893136330, 11.73566716865417, 12.50466445136004, 12.79167228195650, 13.27933591380579, 13.73195896739419, 14.34845176036439, 14.76104822497822
