| 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$ | \( ( 1 - 2 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 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$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 43 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 23 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 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$ | \( ( 1 + 4 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 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$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 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
−9.029973055593751794701534273253, −8.437580779945766317139841544546, −8.010436785697642736041423600877, −7.42819112305891538658843940694, −7.05628584860540280837272529827, −6.24765594036739310828616004723, −5.64808565673135417116444211324, −5.51206865917945044451462215346, −4.79057202611980715265646061167, −4.18535974972689387750755568412, −3.47343971131727371766406808196, −3.21541107080296658914621949913, −2.23362429346088432123663168401, −1.48731869657594409478328408543, 0,
1.48731869657594409478328408543, 2.23362429346088432123663168401, 3.21541107080296658914621949913, 3.47343971131727371766406808196, 4.18535974972689387750755568412, 4.79057202611980715265646061167, 5.51206865917945044451462215346, 5.64808565673135417116444211324, 6.24765594036739310828616004723, 7.05628584860540280837272529827, 7.42819112305891538658843940694, 8.010436785697642736041423600877, 8.437580779945766317139841544546, 9.029973055593751794701534273253
