| L(s) = 1 | + 2-s + 3-s + 4-s + 6-s + 8-s + 9-s − 11-s + 12-s − 2·13-s + 16-s + 18-s − 22-s + 8·23-s + 24-s − 2·26-s + 27-s + 32-s − 33-s + 36-s + 11·37-s − 2·39-s − 44-s + 8·46-s + 47-s + 48-s + 10·49-s − 2·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.301·11-s + 0.288·12-s − 0.554·13-s + 1/4·16-s + 0.235·18-s − 0.213·22-s + 1.66·23-s + 0.204·24-s − 0.392·26-s + 0.192·27-s + 0.176·32-s − 0.174·33-s + 1/6·36-s + 1.80·37-s − 0.320·39-s − 0.150·44-s + 1.17·46-s + 0.145·47-s + 0.144·48-s + 10/7·49-s − 0.277·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 540000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 540000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.887487979\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.887487979\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 23 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 - 3 T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 15 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 + T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 72 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 89 | $C_2^2$ | \( 1 - 52 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
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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
−8.453452049150449289765552693370, −7.920003722434830857641703543890, −7.45958210756944098995706437006, −7.17086949932970788643656593163, −6.76140822423466031960241552159, −5.97012448737822523814049212792, −5.81481564979489075911351761347, −5.04397519806940098357192005856, −4.65973807768922329421292404988, −4.24562425708624212457572371231, −3.57550774201558791021201243480, −2.88157447782265941701565782919, −2.68143607933186552359715607470, −1.86217009487338083890901132695, −0.917332364571166012092665486575,
0.917332364571166012092665486575, 1.86217009487338083890901132695, 2.68143607933186552359715607470, 2.88157447782265941701565782919, 3.57550774201558791021201243480, 4.24562425708624212457572371231, 4.65973807768922329421292404988, 5.04397519806940098357192005856, 5.81481564979489075911351761347, 5.97012448737822523814049212792, 6.76140822423466031960241552159, 7.17086949932970788643656593163, 7.45958210756944098995706437006, 7.920003722434830857641703543890, 8.453452049150449289765552693370
