| L(s) = 1 | − 40·7-s + 420·25-s − 248·31-s − 372·49-s + 120·73-s − 376·79-s + 520·97-s + 2.28e3·103-s + 2.44e3·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 1.74e3·169-s + 173-s − 1.68e4·175-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + ⋯ |
| L(s) = 1 | − 2.15·7-s + 3.35·25-s − 1.43·31-s − 1.08·49-s + 0.192·73-s − 0.535·79-s + 0.544·97-s + 2.18·103-s + 1.83·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s + 0.795·169-s + 0.000439·173-s − 7.25·175-s + 0.000417·179-s + 0.000410·181-s + 0.000378·191-s + 0.000372·193-s + 0.000361·197-s + 0.000356·199-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2981213762\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2981213762\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( ( 1 - 42 p T^{2} + p^{6} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 10 T + p^{3} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 - 1222 T^{2} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2$ | \( ( 1 - 874 T^{2} + p^{6} T^{4} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 + 4194 T^{2} + p^{6} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 + 362 T^{2} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 1806 T^{2} + p^{6} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 12062 T^{2} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 97786 T^{2} + p^{6} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2958 T^{2} + p^{6} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 144934 T^{2} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 4894 T^{2} + p^{6} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - 280114 T^{2} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 127482 T^{2} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 168842 T^{2} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 94646 T^{2} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{3} T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 94 T + p^{3} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 694134 T^{2} + p^{6} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 846738 T^{2} + p^{6} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 130 T + p^{3} T^{2} )^{4} \) |
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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
−8.182955577806507176805367752384, −7.973105939005372437240550521558, −7.47797754408745442202415847960, −7.36837078979486428617197039598, −6.87359066905063971362898920330, −6.73590517830974794752686196014, −6.73018309164840051319366687763, −6.35275359434611735562757618653, −6.17949799467147215748418856336, −5.71787271307614744210991680586, −5.47208158849970056064865173978, −5.23983854672807806249488239578, −4.74174444524037927031743085510, −4.63733058652809752898203247451, −4.36404430933712176673413138925, −3.65774777900274509571381063927, −3.49911373794932381001709001421, −3.35143552432572228030071450784, −2.84771545662906970637974875852, −2.83972618018538893084396259883, −2.22932242136234250296507859662, −1.72726583032781484984080730914, −1.19651747452334224848339462325, −0.76095503809034163300793139936, −0.11598757161119102119435961137,
0.11598757161119102119435961137, 0.76095503809034163300793139936, 1.19651747452334224848339462325, 1.72726583032781484984080730914, 2.22932242136234250296507859662, 2.83972618018538893084396259883, 2.84771545662906970637974875852, 3.35143552432572228030071450784, 3.49911373794932381001709001421, 3.65774777900274509571381063927, 4.36404430933712176673413138925, 4.63733058652809752898203247451, 4.74174444524037927031743085510, 5.23983854672807806249488239578, 5.47208158849970056064865173978, 5.71787271307614744210991680586, 6.17949799467147215748418856336, 6.35275359434611735562757618653, 6.73018309164840051319366687763, 6.73590517830974794752686196014, 6.87359066905063971362898920330, 7.36837078979486428617197039598, 7.47797754408745442202415847960, 7.973105939005372437240550521558, 8.182955577806507176805367752384
