| L(s) = 1 | + 36·5-s − 30·9-s + 356·13-s − 252·17-s − 278·25-s − 2.84e3·29-s + 1.06e3·37-s + 324·41-s − 1.08e3·45-s + 4.03e3·49-s + 1.18e3·53-s + 1.25e3·61-s + 1.28e4·65-s − 1.33e4·73-s − 5.66e3·81-s − 9.07e3·85-s + 1.64e4·89-s − 3.19e3·97-s − 9.18e3·101-s + 1.87e4·109-s + 3.77e4·113-s − 1.06e4·117-s + 1.37e4·121-s − 4.41e4·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | + 1.43·5-s − 0.370·9-s + 2.10·13-s − 0.871·17-s − 0.444·25-s − 3.38·29-s + 0.774·37-s + 0.192·41-s − 0.533·45-s + 1.68·49-s + 0.422·53-s + 0.336·61-s + 3.03·65-s − 2.50·73-s − 0.862·81-s − 1.25·85-s + 2.07·89-s − 0.339·97-s − 0.899·101-s + 1.58·109-s + 2.95·113-s − 0.780·117-s + 0.937·121-s − 2.82·125-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.591892918\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.591892918\) |
| \(L(3)\) |
|
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 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + 10 p T^{2} + p^{8} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 18 T + p^{4} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 94 T + p^{4} T^{2} )( 1 + 94 T + p^{4} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 13730 T^{2} + p^{8} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 178 T + p^{4} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 126 T + p^{4} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 99170 T^{2} + p^{8} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 190 T^{2} + p^{8} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 1422 T + p^{4} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1736450 T^{2} + p^{8} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 530 T + p^{4} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 162 T + p^{4} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 4471970 T^{2} + p^{8} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 2433406 T^{2} + p^{8} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 594 T + p^{4} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 18620450 T^{2} + p^{8} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 626 T + p^{4} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 39103970 T^{2} + p^{8} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 8958526 T^{2} + p^{8} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 6686 T + p^{4} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 75980162 T^{2} + p^{8} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 73625954 T^{2} + p^{8} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 8226 T + p^{4} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 1598 T + p^{4} T^{2} )^{2} \) |
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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
−18.49403857721936048837836211054, −18.06836023437643585814049186767, −17.44936760526088162295216829859, −16.84154656385198925609835861715, −16.15714647425047230237525206001, −15.36696591063106557177972043046, −14.67508805471552722410182275946, −13.64759522673011492851164922547, −13.46465136652306531144306691342, −12.89920524090815261321697102276, −11.46607753553729154749631859307, −11.08207130663272797544744924450, −10.12250259294611252202556742692, −9.217531861594741111994238977521, −8.715598346365770058092297180882, −7.41165828246141512397470417158, −5.94309129791928061554367690148, −5.83299655036539627994944390209, −3.86456321740832332010452948237, −1.92229369019212028723216941492,
1.92229369019212028723216941492, 3.86456321740832332010452948237, 5.83299655036539627994944390209, 5.94309129791928061554367690148, 7.41165828246141512397470417158, 8.715598346365770058092297180882, 9.217531861594741111994238977521, 10.12250259294611252202556742692, 11.08207130663272797544744924450, 11.46607753553729154749631859307, 12.89920524090815261321697102276, 13.46465136652306531144306691342, 13.64759522673011492851164922547, 14.67508805471552722410182275946, 15.36696591063106557177972043046, 16.15714647425047230237525206001, 16.84154656385198925609835861715, 17.44936760526088162295216829859, 18.06836023437643585814049186767, 18.49403857721936048837836211054
