| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s − 2·9-s − 6·11-s − 12-s + 8·13-s + 16-s − 2·18-s − 6·22-s − 12·23-s − 24-s + 8·26-s + 5·27-s + 32-s + 6·33-s − 2·36-s − 4·37-s − 8·39-s − 6·44-s − 12·46-s − 24·47-s − 48-s − 10·49-s + 8·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.408·6-s + 0.353·8-s − 2/3·9-s − 1.80·11-s − 0.288·12-s + 2.21·13-s + 1/4·16-s − 0.471·18-s − 1.27·22-s − 2.50·23-s − 0.204·24-s + 1.56·26-s + 0.962·27-s + 0.176·32-s + 1.04·33-s − 1/3·36-s − 0.657·37-s − 1.28·39-s − 0.904·44-s − 1.76·46-s − 3.50·47-s − 0.144·48-s − 1.42·49-s + 1.10·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 180000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 180000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_1$ | \( 1 - T \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.618275843797833804091344971318, −8.311607052372601690738905070254, −8.071729218562685355881418106440, −7.60180604693367715686676939784, −6.59457926685779196027153945229, −6.35306147256574072432179047412, −5.99247695350663513053687244638, −5.34945783064988058695168525193, −5.10038235657027705713371559541, −4.36318204480909307645122292983, −3.55259950777558712846266054037, −3.29699975126484092788987476482, −2.36548560349712652287575039514, −1.59113933029870209082752356824, 0,
1.59113933029870209082752356824, 2.36548560349712652287575039514, 3.29699975126484092788987476482, 3.55259950777558712846266054037, 4.36318204480909307645122292983, 5.10038235657027705713371559541, 5.34945783064988058695168525193, 5.99247695350663513053687244638, 6.35306147256574072432179047412, 6.59457926685779196027153945229, 7.60180604693367715686676939784, 8.071729218562685355881418106440, 8.311607052372601690738905070254, 8.618275843797833804091344971318
