| L(s) = 1 | − 2·2-s + 8·8-s − 4·13-s − 16·16-s + 20·17-s + 8·26-s + 52·29-s − 40·34-s − 52·37-s − 116·41-s + 50·49-s − 148·53-s − 104·58-s + 52·61-s + 64·64-s + 92·73-s + 104·74-s + 232·82-s − 164·89-s − 4·97-s − 100·98-s + 148·101-s − 32·104-s + 296·106-s − 92·109-s − 220·113-s + 194·121-s + ⋯ |
| L(s) = 1 | − 2-s + 8-s − 0.307·13-s − 16-s + 1.17·17-s + 4/13·26-s + 1.79·29-s − 1.17·34-s − 1.40·37-s − 2.82·41-s + 1.02·49-s − 2.79·53-s − 1.79·58-s + 0.852·61-s + 64-s + 1.26·73-s + 1.40·74-s + 2.82·82-s − 1.84·89-s − 0.0412·97-s − 1.02·98-s + 1.46·101-s − 0.307·104-s + 2.79·106-s − 0.844·109-s − 1.94·113-s + 1.60·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.8405744337\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8405744337\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p T + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 50 T^{2} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 194 T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 290 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1874 T^{2} + p^{4} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 26 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 1346 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 382 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 74 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 1150 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 26 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 8930 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 46 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 1390 T^{2} + p^{4} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 11426 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
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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
−10.10121736211881363732489087483, −9.801363894747231256086052369534, −9.276319658595028567422831167300, −8.869219056415521037590423395857, −8.426193407961653379349674987471, −7.990747665594427114415226153817, −7.951735487941129261012602524697, −7.15904640976172747630210728069, −6.68965475963216073109836709197, −6.65475508742747312508217369889, −5.62162409105290670025508667788, −5.39107564862707053524326507204, −4.67716430119025593109464936858, −4.57737962263966161327203843505, −3.53769149839324454120994755918, −3.36038304632522444304701878812, −2.53214787722504622801103474391, −1.72820330924971449169097707782, −1.25159932223937017415921956428, −0.39056070180863128086165661181,
0.39056070180863128086165661181, 1.25159932223937017415921956428, 1.72820330924971449169097707782, 2.53214787722504622801103474391, 3.36038304632522444304701878812, 3.53769149839324454120994755918, 4.57737962263966161327203843505, 4.67716430119025593109464936858, 5.39107564862707053524326507204, 5.62162409105290670025508667788, 6.65475508742747312508217369889, 6.68965475963216073109836709197, 7.15904640976172747630210728069, 7.951735487941129261012602524697, 7.990747665594427114415226153817, 8.426193407961653379349674987471, 8.869219056415521037590423395857, 9.276319658595028567422831167300, 9.801363894747231256086052369534, 10.10121736211881363732489087483
