L(s) = 1 | + 3-s − 2·4-s − 7-s − 2·9-s − 11-s − 2·12-s + 13-s + 4·16-s + 17-s + 2·19-s − 21-s + 23-s − 5·27-s + 2·28-s + 7·29-s + 4·31-s − 33-s + 4·36-s + 8·37-s + 39-s − 6·41-s − 8·43-s + 2·44-s − 7·47-s + 4·48-s + 49-s + 51-s + ⋯ |
L(s) = 1 | + 0.577·3-s − 4-s − 0.377·7-s − 2/3·9-s − 0.301·11-s − 0.577·12-s + 0.277·13-s + 16-s + 0.242·17-s + 0.458·19-s − 0.218·21-s + 0.208·23-s − 0.962·27-s + 0.377·28-s + 1.29·29-s + 0.718·31-s − 0.174·33-s + 2/3·36-s + 1.31·37-s + 0.160·39-s − 0.937·41-s − 1.21·43-s + 0.301·44-s − 1.02·47-s + 0.577·48-s + 1/7·49-s + 0.140·51-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 + p T^{2} \) |
| 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 15 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−8.249051193477960890115594197464, −7.62779720098297255017666351116, −6.53454340706693236827806403764, −5.79992371633617361455709349591, −5.00396804939761087566103974500, −4.26334706188802962808282230504, −3.24057586493752760359257557209, −2.82943919034522223313588838065, −1.31252271658192077335908136807, 0,
1.31252271658192077335908136807, 2.82943919034522223313588838065, 3.24057586493752760359257557209, 4.26334706188802962808282230504, 5.00396804939761087566103974500, 5.79992371633617361455709349591, 6.53454340706693236827806403764, 7.62779720098297255017666351116, 8.249051193477960890115594197464