L(s) = 1 | + 8·17-s + 16·29-s + 8·37-s − 16·41-s − 8·49-s + 16·53-s + 12·61-s + 8·73-s − 9·81-s + 4·89-s − 8·97-s + 16·101-s − 20·109-s − 32·113-s + 2·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 26·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | + 1.94·17-s + 2.97·29-s + 1.31·37-s − 2.49·41-s − 8/7·49-s + 2.19·53-s + 1.53·61-s + 0.936·73-s − 81-s + 0.423·89-s − 0.812·97-s + 1.59·101-s − 1.91·109-s − 3.01·113-s + 2/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2·169-s + 0.0760·173-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.760348995\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.760348995\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 32 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 56 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 94 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 128 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 160 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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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
−8.105998031284016012497066292006, −7.954799601634344455155122917612, −7.62890890674004115222922651125, −7.00459671187641933544010446899, −6.77269287523866598830353484878, −6.54654407684128169298835337048, −6.17014640255449720284810565530, −5.57076527858526272859525031920, −5.26971051456388481707688347329, −5.22566468890747090335757145352, −4.51418238034336033166893020914, −4.30201026754401379423658881678, −3.75867801212458246566011520385, −3.38938182326884993030457394286, −2.86285572517758780791193627144, −2.77399758790051549157289527118, −2.06792493534089542560156246395, −1.48890073033772461153367357040, −0.986873156739337986796819432605, −0.58660412315674140250621442611,
0.58660412315674140250621442611, 0.986873156739337986796819432605, 1.48890073033772461153367357040, 2.06792493534089542560156246395, 2.77399758790051549157289527118, 2.86285572517758780791193627144, 3.38938182326884993030457394286, 3.75867801212458246566011520385, 4.30201026754401379423658881678, 4.51418238034336033166893020914, 5.22566468890747090335757145352, 5.26971051456388481707688347329, 5.57076527858526272859525031920, 6.17014640255449720284810565530, 6.54654407684128169298835337048, 6.77269287523866598830353484878, 7.00459671187641933544010446899, 7.62890890674004115222922651125, 7.954799601634344455155122917612, 8.105998031284016012497066292006