| L(s) = 1 | − 3-s − 2·4-s − 5-s − 2·9-s + 2·12-s − 13-s + 15-s + 4·16-s + 2·20-s + 25-s + 5·27-s − 14·31-s + 4·36-s + 8·37-s + 39-s + 9·41-s + 11·43-s + 2·45-s − 4·48-s + 2·49-s + 2·52-s − 3·53-s − 2·60-s − 8·64-s + 65-s + 17·67-s − 9·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 4-s − 0.447·5-s − 2/3·9-s + 0.577·12-s − 0.277·13-s + 0.258·15-s + 16-s + 0.447·20-s + 1/5·25-s + 0.962·27-s − 2.51·31-s + 2/3·36-s + 1.31·37-s + 0.160·39-s + 1.40·41-s + 1.67·43-s + 0.298·45-s − 0.577·48-s + 2/7·49-s + 0.277·52-s − 0.412·53-s − 0.258·60-s − 64-s + 0.124·65-s + 2.07·67-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_2$ | \( 1 + p T^{2} \) |
| 3 | $C_2$ | \( 1 + T + p T^{2} \) |
| 5 | $C_1$ | \( 1 + T \) |
| good | 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 8 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 29 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 + 4 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2^2$ | \( 1 + 136 T^{2} + p^{2} T^{4} \) |
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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
−8.933704118300936802591523900027, −8.432953852457324344679654943509, −7.76492047294845852924304826356, −7.50792686685256150029334773001, −6.99299729246842230861448013078, −6.16613662934134908604003916356, −5.71806650566000179487289720171, −5.48323312196914960455202802870, −4.76738470375839414337236762761, −4.25906761632603670109845527152, −3.79996533598904598436090270715, −3.06483335293129371128757025033, −2.33517872487363433910125537907, −1.04459128719907683856615859851, 0,
1.04459128719907683856615859851, 2.33517872487363433910125537907, 3.06483335293129371128757025033, 3.79996533598904598436090270715, 4.25906761632603670109845527152, 4.76738470375839414337236762761, 5.48323312196914960455202802870, 5.71806650566000179487289720171, 6.16613662934134908604003916356, 6.99299729246842230861448013078, 7.50792686685256150029334773001, 7.76492047294845852924304826356, 8.432953852457324344679654943509, 8.933704118300936802591523900027
