| L(s) = 1 | − 24·7-s + 14·9-s − 36·25-s + 88·31-s + 164·49-s − 336·63-s − 328·73-s − 40·79-s + 115·81-s − 376·97-s − 536·103-s − 420·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 476·169-s + 173-s + 864·175-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 3.42·7-s + 14/9·9-s − 1.43·25-s + 2.83·31-s + 3.34·49-s − 5.33·63-s − 4.49·73-s − 0.506·79-s + 1.41·81-s − 3.87·97-s − 5.20·103-s − 3.47·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 0.00598·167-s + 2.81·169-s + 0.00578·173-s + 4.93·175-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{32} \cdot 3^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.5551945573\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5551945573\) |
| \(L(2)\) |
|
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 | $C_2^2$ | \( 1 - 14 T^{2} + p^{4} T^{4} \) |
| good | 5 | $C_2^2$ | \( ( 1 + 18 T^{2} + p^{4} T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 6 T + p^{2} T^{2} )^{4} \) |
| 11 | $C_2^2$ | \( ( 1 + 210 T^{2} + p^{4} T^{4} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 24 T + p^{2} T^{2} )^{2}( 1 + 24 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( ( 1 - 66 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^2$ | \( ( 1 - 718 T^{2} + p^{4} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 - 930 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 + 1394 T^{2} + p^{4} T^{4} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 22 T + p^{2} T^{2} )^{4} \) |
| 37 | $C_2^2$ | \( ( 1 - 2702 T^{2} + p^{4} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 - 2210 T^{2} + p^{4} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + 3026 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + 190 T^{2} + p^{4} T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 + 1746 T^{2} + p^{4} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 1554 T^{2} + p^{4} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 46 T^{2} + p^{4} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 8974 T^{2} + p^{4} T^{4} )^{2} \) |
| 71 | $C_2^2$ | \( ( 1 + 5406 T^{2} + p^{4} T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 82 T + p^{2} T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + 10 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + 8370 T^{2} + p^{4} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 - 14690 T^{2} + p^{4} T^{4} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 94 T + p^{2} T^{2} )^{4} \) |
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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
−6.83607013004234900522574519182, −6.81672755673568507080726123486, −6.79738202810207090298406923215, −6.68433079768572424276658374055, −6.31030830261777643873984525689, −6.24563780059640552444061913941, −5.89296667811116547307356361554, −5.49903549427804157577999915932, −5.39816111247757074178685494784, −5.32130397745471443043517603631, −4.56344913750974262061837095086, −4.39128476772537536874481324836, −4.18903336733795748348759670342, −4.16682248739447683973938964032, −3.77560825867312129419565300719, −3.46215419941368444778660995551, −2.98000434247607352247063012204, −2.89422824708582426382274662847, −2.83047362663289718757459606306, −2.54518158186459026504426355315, −1.74485353870386351232963488559, −1.47641504114003126758450366172, −1.30717438850800961820420181827, −0.51522788665765764518583575168, −0.18203311097186505170605348271,
0.18203311097186505170605348271, 0.51522788665765764518583575168, 1.30717438850800961820420181827, 1.47641504114003126758450366172, 1.74485353870386351232963488559, 2.54518158186459026504426355315, 2.83047362663289718757459606306, 2.89422824708582426382274662847, 2.98000434247607352247063012204, 3.46215419941368444778660995551, 3.77560825867312129419565300719, 4.16682248739447683973938964032, 4.18903336733795748348759670342, 4.39128476772537536874481324836, 4.56344913750974262061837095086, 5.32130397745471443043517603631, 5.39816111247757074178685494784, 5.49903549427804157577999915932, 5.89296667811116547307356361554, 6.24563780059640552444061913941, 6.31030830261777643873984525689, 6.68433079768572424276658374055, 6.79738202810207090298406923215, 6.81672755673568507080726123486, 6.83607013004234900522574519182
