| L(s) = 1 | − 2-s + 4·7-s + 8-s + 3·11-s + 4·13-s − 4·14-s − 16-s + 6·17-s − 8·19-s − 3·22-s − 6·23-s − 4·26-s − 6·29-s − 8·31-s − 6·34-s + 16·37-s + 8·38-s − 6·41-s + 43-s + 6·46-s + 12·47-s + 7·49-s + 4·56-s + 6·58-s − 9·59-s − 8·61-s + 8·62-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.51·7-s + 0.353·8-s + 0.904·11-s + 1.10·13-s − 1.06·14-s − 1/4·16-s + 1.45·17-s − 1.83·19-s − 0.639·22-s − 1.25·23-s − 0.784·26-s − 1.11·29-s − 1.43·31-s − 1.02·34-s + 2.63·37-s + 1.29·38-s − 0.937·41-s + 0.152·43-s + 0.884·46-s + 1.75·47-s + 49-s + 0.534·56-s + 0.787·58-s − 1.17·59-s − 1.02·61-s + 1.01·62-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.940042444\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.940042444\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 3 T - 2 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4 T + 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 12 T + 97 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 8 T - 15 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 9 T - 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 7 T - 48 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
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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
−9.565890093667716301903475017030, −9.364794555117639158792231120382, −9.063108532789975321992806807107, −8.567942455973145494780753780013, −8.067930272505642360470463524102, −7.935716181500996444434983568400, −7.70506426291676780616313295019, −7.08272488617641963911136490289, −6.39502493280884012184012445830, −6.25106938629904300476293495052, −5.50703514589620116615725634470, −5.47544382176643890054941498285, −4.54419795569129153228288426921, −4.26832738104199521390960582454, −3.81218424781114364492048393479, −3.44403331388401061490807130214, −2.34054753841674087076194131474, −1.91052884048288499665194001910, −1.41957958555812011113405126233, −0.70414592412554274768584317302,
0.70414592412554274768584317302, 1.41957958555812011113405126233, 1.91052884048288499665194001910, 2.34054753841674087076194131474, 3.44403331388401061490807130214, 3.81218424781114364492048393479, 4.26832738104199521390960582454, 4.54419795569129153228288426921, 5.47544382176643890054941498285, 5.50703514589620116615725634470, 6.25106938629904300476293495052, 6.39502493280884012184012445830, 7.08272488617641963911136490289, 7.70506426291676780616313295019, 7.935716181500996444434983568400, 8.067930272505642360470463524102, 8.567942455973145494780753780013, 9.063108532789975321992806807107, 9.364794555117639158792231120382, 9.565890093667716301903475017030
