L(s) = 1 | − 2-s − 4-s + 3·8-s − 5·11-s + 5·13-s − 16-s − 4·17-s − 2·19-s + 5·22-s + 3·23-s − 5·26-s − 10·29-s + 6·31-s − 5·32-s + 4·34-s − 5·37-s + 2·38-s − 10·41-s − 10·43-s + 5·44-s − 3·46-s + 5·47-s − 7·49-s − 5·52-s + 2·53-s + 10·58-s − 5·59-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.50·11-s + 1.38·13-s − 1/4·16-s − 0.970·17-s − 0.458·19-s + 1.06·22-s + 0.625·23-s − 0.980·26-s − 1.85·29-s + 1.07·31-s − 0.883·32-s + 0.685·34-s − 0.821·37-s + 0.324·38-s − 1.56·41-s − 1.52·43-s + 0.753·44-s − 0.442·46-s + 0.729·47-s − 49-s − 0.693·52-s + 0.274·53-s + 1.31·58-s − 0.650·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
good | 2 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + 11 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 5 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−10.12687139605402871610685180919, −9.028285709832855333537102502485, −8.470229778473156859055797250625, −7.72941864643843111924093742058, −6.65161085258324315501961849103, −5.45093501870232524860958154077, −4.56767724494133603201732869283, −3.35142569623968027183576325655, −1.76125584480235740498550240381, 0,
1.76125584480235740498550240381, 3.35142569623968027183576325655, 4.56767724494133603201732869283, 5.45093501870232524860958154077, 6.65161085258324315501961849103, 7.72941864643843111924093742058, 8.470229778473156859055797250625, 9.028285709832855333537102502485, 10.12687139605402871610685180919