| L(s) = 1 | + 2·5-s − 24·7-s + 44·11-s − 22·13-s + 100·17-s + 88·19-s + 56·23-s + 125·25-s − 198·29-s + 160·31-s − 48·35-s − 324·37-s + 198·41-s − 52·43-s − 528·47-s + 343·49-s − 484·53-s + 88·55-s + 668·59-s − 550·61-s − 44·65-s − 188·67-s + 1.45e3·71-s + 308·73-s − 1.05e3·77-s + 656·79-s − 236·83-s + ⋯ |
| L(s) = 1 | + 0.178·5-s − 1.29·7-s + 1.20·11-s − 0.469·13-s + 1.42·17-s + 1.06·19-s + 0.507·23-s + 25-s − 1.26·29-s + 0.926·31-s − 0.231·35-s − 1.43·37-s + 0.754·41-s − 0.184·43-s − 1.63·47-s + 49-s − 1.25·53-s + 0.215·55-s + 1.47·59-s − 1.15·61-s − 0.0839·65-s − 0.342·67-s + 2.43·71-s + 0.493·73-s − 1.56·77-s + 0.934·79-s − 0.312·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 419904 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.675283611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.675283611\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - 121 T^{2} - 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 24 T + 233 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 4 p T + 5 p^{2} T^{2} - 4 p^{4} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 22 T - 1713 T^{2} + 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 50 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 44 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 56 T - 9031 T^{2} - 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 198 T + 14815 T^{2} + 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 160 T - 4191 T^{2} - 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 162 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 198 T - 29717 T^{2} - 198 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 52 T - 76803 T^{2} + 52 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 528 T + 174961 T^{2} + 528 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 242 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 668 T + 240845 T^{2} - 668 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 550 T + 75519 T^{2} + 550 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 188 T - 265419 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 728 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 154 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 656 T - 62703 T^{2} - 656 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 236 T - 516091 T^{2} + 236 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 714 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 478 T - 684189 T^{2} - 478 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−10.30740204895596792771543032545, −9.705442943488627302536992430161, −9.517377452981288729040101594810, −9.353711418763977708710569056692, −8.777908651249621726125500917504, −8.099824969658361213689297638015, −7.79432931021386428096195479283, −7.15517390852697276661644232836, −6.64726285703222029146915871993, −6.61395897162721266958476941329, −5.88129886873754225330631503353, −5.32388264196708886643441223009, −5.04853774915963060023383305051, −4.26190156154995160052998218185, −3.43347335047732781255646721776, −3.42584620501000330047134401024, −2.80415422233293758430000314879, −1.86568405869479346196455974915, −1.15308100433438079451612575080, −0.52905647721533198470473183562,
0.52905647721533198470473183562, 1.15308100433438079451612575080, 1.86568405869479346196455974915, 2.80415422233293758430000314879, 3.42584620501000330047134401024, 3.43347335047732781255646721776, 4.26190156154995160052998218185, 5.04853774915963060023383305051, 5.32388264196708886643441223009, 5.88129886873754225330631503353, 6.61395897162721266958476941329, 6.64726285703222029146915871993, 7.15517390852697276661644232836, 7.79432931021386428096195479283, 8.099824969658361213689297638015, 8.777908651249621726125500917504, 9.353711418763977708710569056692, 9.517377452981288729040101594810, 9.705442943488627302536992430161, 10.30740204895596792771543032545
