| L(s) = 1 | − 2·2-s + 3·4-s + 2·5-s − 2·7-s − 4·8-s − 4·10-s + 2·11-s + 4·13-s + 4·14-s + 5·16-s − 4·17-s + 6·20-s − 4·22-s + 3·25-s − 8·26-s − 6·28-s − 12·29-s − 6·32-s + 8·34-s − 4·35-s + 12·37-s − 8·40-s − 4·41-s − 8·43-s + 6·44-s + 16·47-s + 3·49-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 0.894·5-s − 0.755·7-s − 1.41·8-s − 1.26·10-s + 0.603·11-s + 1.10·13-s + 1.06·14-s + 5/4·16-s − 0.970·17-s + 1.34·20-s − 0.852·22-s + 3/5·25-s − 1.56·26-s − 1.13·28-s − 2.22·29-s − 1.06·32-s + 1.37·34-s − 0.676·35-s + 1.97·37-s − 1.26·40-s − 0.624·41-s − 1.21·43-s + 0.904·44-s + 2.33·47-s + 3/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48024900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.986482339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.986482339\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_4$ | \( 1 + 4 T + 6 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 86 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 102 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 70 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 4 T + 38 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 4 T - 2 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 8 T + 22 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 4 T + 118 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16 T + 214 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 16 T + 198 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 12 T + 182 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 190 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−8.135456092379863405661776558643, −7.958396207286525661007157354510, −7.37182532142529838037535128381, −7.06850411045009298588425919310, −6.73614129240646894274607636517, −6.59960604441439169607713175404, −6.04589132637168113730732646337, −5.89395603376242765595164866473, −5.44264740363302967203712321731, −5.27222859816706035626839762504, −4.26693469931048318465699901979, −4.23329283308185361339478924468, −3.53394846711672956292660180291, −3.45211045655002986373715699057, −2.62640663004094932202407881308, −2.43157830439672378791915859189, −1.85113744344016151172689046530, −1.64138778229613581179116197128, −0.75523982962365721079831247592, −0.61308429793714310347005512054,
0.61308429793714310347005512054, 0.75523982962365721079831247592, 1.64138778229613581179116197128, 1.85113744344016151172689046530, 2.43157830439672378791915859189, 2.62640663004094932202407881308, 3.45211045655002986373715699057, 3.53394846711672956292660180291, 4.23329283308185361339478924468, 4.26693469931048318465699901979, 5.27222859816706035626839762504, 5.44264740363302967203712321731, 5.89395603376242765595164866473, 6.04589132637168113730732646337, 6.59960604441439169607713175404, 6.73614129240646894274607636517, 7.06850411045009298588425919310, 7.37182532142529838037535128381, 7.958396207286525661007157354510, 8.135456092379863405661776558643
