| L(s) = 1 | + 2·9-s + 20·17-s + 12·41-s − 12·49-s − 60·73-s − 15·81-s + 4·89-s − 40·97-s + 60·113-s + 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 40·153-s + 157-s + 163-s + 167-s − 28·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + ⋯ |
| L(s) = 1 | + 2/3·9-s + 4.85·17-s + 1.87·41-s − 1.71·49-s − 7.02·73-s − 5/3·81-s + 0.423·89-s − 4.06·97-s + 5.64·113-s + 3.09·121-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 + 3.23·153-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s − 2.15·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.933380048\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.933380048\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - T^{2} + p^{2} T^{4} )^{2} \) |
| 7 | $C_2^2$ | \( ( 1 + 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2^2$ | \( ( 1 - 17 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{4} \) |
| 19 | $C_2^2$ | \( ( 1 - 33 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 14 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 18 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 34 T^{2} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{4} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 86 T^{2} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 54 T^{2} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - 38 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 9 T^{2} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{4} \) |
| 79 | $C_2^2$ | \( ( 1 - 42 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2$ | \( ( 1 - 121 T^{2} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−5.99950637814402003534263259890, −5.96016834983085581476109390383, −5.79677986038035335978775471115, −5.60711551725294787070819179736, −5.49282252838308795576768562759, −5.10717591652989919941164985875, −4.90696250564629571546894774338, −4.75523619421423959895978080407, −4.39879693986184581989379264049, −4.33872849421940235836809109840, −4.15618811157245201920897066324, −3.83133781389755788404528064438, −3.47622982722415431054841147769, −3.43070761547423316832095447571, −3.12250350340757572555393690304, −3.05022361619347850846758734255, −2.72694223950730905013283491281, −2.68952731779988300721215996494, −2.15258750165526804654800071894, −1.73106325099570893526123908111, −1.42715603677567071817253151567, −1.32162224478471208485192027950, −1.29827131767529395022721206120, −0.68537427364798018976735953965, −0.39138909433923943845512505527,
0.39138909433923943845512505527, 0.68537427364798018976735953965, 1.29827131767529395022721206120, 1.32162224478471208485192027950, 1.42715603677567071817253151567, 1.73106325099570893526123908111, 2.15258750165526804654800071894, 2.68952731779988300721215996494, 2.72694223950730905013283491281, 3.05022361619347850846758734255, 3.12250350340757572555393690304, 3.43070761547423316832095447571, 3.47622982722415431054841147769, 3.83133781389755788404528064438, 4.15618811157245201920897066324, 4.33872849421940235836809109840, 4.39879693986184581989379264049, 4.75523619421423959895978080407, 4.90696250564629571546894774338, 5.10717591652989919941164985875, 5.49282252838308795576768562759, 5.60711551725294787070819179736, 5.79677986038035335978775471115, 5.96016834983085581476109390383, 5.99950637814402003534263259890
