| L(s) = 1 | + 2-s − 3-s − 5-s − 6-s − 8-s − 10-s + 4·13-s + 15-s − 16-s + 6·17-s − 8·19-s + 24-s + 4·26-s + 27-s + 12·29-s + 30-s + 4·31-s + 6·34-s + 10·37-s − 8·38-s − 4·39-s + 40-s − 12·41-s − 8·43-s + 48-s − 6·51-s + 6·53-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 0.447·5-s − 0.408·6-s − 0.353·8-s − 0.316·10-s + 1.10·13-s + 0.258·15-s − 1/4·16-s + 1.45·17-s − 1.83·19-s + 0.204·24-s + 0.784·26-s + 0.192·27-s + 2.22·29-s + 0.182·30-s + 0.718·31-s + 1.02·34-s + 1.64·37-s − 1.29·38-s − 0.640·39-s + 0.158·40-s − 1.87·41-s − 1.21·43-s + 0.144·48-s − 0.840·51-s + 0.824·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.373490270\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.373490270\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 10 T + 39 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
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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
−9.791434880902513186132208580318, −9.449215582184132038465277622427, −8.614272226404186525909055427698, −8.430246770546019188023280837039, −8.121710921245702784045025082274, −8.004426382159837622076155639792, −6.87063015451002989658246319533, −6.86540020921682057705218605043, −6.33430221595517317096181717659, −6.13443791549414235761296963655, −5.46613137025875440917645636038, −5.09490089267121241179895457275, −4.78477328747497908082097298498, −4.16254106534565479744723962532, −3.71077312113999497484978498745, −3.50787849130044246930199504118, −2.67786113457138663816340637913, −2.25599956164279389955355448938, −1.19960498465657348157022178252, −0.65743226668061380058924481398,
0.65743226668061380058924481398, 1.19960498465657348157022178252, 2.25599956164279389955355448938, 2.67786113457138663816340637913, 3.50787849130044246930199504118, 3.71077312113999497484978498745, 4.16254106534565479744723962532, 4.78477328747497908082097298498, 5.09490089267121241179895457275, 5.46613137025875440917645636038, 6.13443791549414235761296963655, 6.33430221595517317096181717659, 6.86540020921682057705218605043, 6.87063015451002989658246319533, 8.004426382159837622076155639792, 8.121710921245702784045025082274, 8.430246770546019188023280837039, 8.614272226404186525909055427698, 9.449215582184132038465277622427, 9.791434880902513186132208580318
