| L(s) = 1 | − 12·3-s + 54·9-s + 23·16-s − 168·29-s − 276·48-s − 96·53-s + 120·61-s + 216·79-s − 1.21e3·81-s + 2.01e3·87-s + 456·107-s − 504·113-s + 127-s + 131-s + 137-s + 139-s + 1.24e3·144-s + 149-s + 151-s + 157-s + 1.15e3·159-s + 163-s + 167-s + 173-s + 179-s + 181-s − 1.44e3·183-s + ⋯ |
| L(s) = 1 | − 4·3-s + 6·9-s + 1.43·16-s − 5.79·29-s − 5.75·48-s − 1.81·53-s + 1.96·61-s + 2.73·79-s − 15·81-s + 23.1·87-s + 4.26·107-s − 4.46·113-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 69/8·144-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 7.24·159-s + 0.00613·163-s + 0.00598·167-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s − 7.86·183-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(13^{8}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.2541972445\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2541972445\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 13 | | \( 1 \) |
| good | 2 | $C_2^3$ | \( 1 - 23 T^{4} + p^{8} T^{8} \) |
| 3 | $C_2$ | \( ( 1 + p T + p^{2} T^{2} )^{4} \) |
| 5 | $C_2^3$ | \( 1 + 271 T^{4} + p^{8} T^{8} \) |
| 7 | $C_2^3$ | \( 1 - 4801 T^{4} + p^{8} T^{8} \) |
| 11 | $C_2^3$ | \( 1 + 82 p^{2} T^{4} + p^{8} T^{8} \) |
| 17 | $C_2^2$ | \( ( 1 - 569 T^{2} + p^{4} T^{4} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - 154366 T^{4} + p^{8} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 914 T^{2} + p^{4} T^{4} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 42 T + p^{2} T^{2} )^{4} \) |
| 31 | $C_2^3$ | \( 1 - 1732798 T^{4} + p^{8} T^{8} \) |
| 37 | $C_2^3$ | \( 1 - 3679153 T^{4} + p^{8} T^{8} \) |
| 41 | $C_2^3$ | \( 1 - 4197086 T^{4} + p^{8} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 - 1297 T^{2} + p^{4} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 + 9662287 T^{4} + p^{8} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 24 T + p^{2} T^{2} )^{4} \) |
| 59 | $C_2^3$ | \( 1 - 7045406 T^{4} + p^{8} T^{8} \) |
| 61 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{4} \) |
| 67 | $C_2^3$ | \( 1 + 14368994 T^{4} + p^{8} T^{8} \) |
| 71 | $C_2^3$ | \( 1 + 40245487 T^{4} + p^{8} T^{8} \) |
| 73 | $C_2^3$ | \( 1 + 25540994 T^{4} + p^{8} T^{8} \) |
| 79 | $C_2$ | \( ( 1 - 54 T + p^{2} T^{2} )^{4} \) |
| 83 | $C_2^3$ | \( 1 + 40154242 T^{4} + p^{8} T^{8} \) |
| 89 | $C_2^3$ | \( 1 + 10424482 T^{4} + p^{8} T^{8} \) |
| 97 | $C_2^3$ | \( 1 + 162311522 T^{4} + p^{8} T^{8} \) |
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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
−9.341850737456151998212655471552, −8.805857313211130631884419535493, −8.708148978305236828187232442431, −8.207055330665520210720896398354, −7.982852870821903329330192327278, −7.51891451909267780576667476718, −7.44521390189786916926397691729, −7.03095525451888981842280536482, −6.61332342840504900454469836819, −6.32901422867944884315413789571, −6.25983510352376037865786824071, −5.71759443496550371006168742915, −5.64795850130428998962826749704, −5.60616039871882773883519767843, −5.21341032749765676133468647104, −4.99055943274169024918634350643, −4.79416373267360741953969097080, −3.92066980835038460154751971117, −3.91218329087799270974986489570, −3.29982501017726773349076263126, −2.92801790437626255193970967463, −2.04687334041393337655454193340, −1.55623987076872935983950675627, −0.50687259975048986983650056356, −0.48741489902242836634800069835,
0.48741489902242836634800069835, 0.50687259975048986983650056356, 1.55623987076872935983950675627, 2.04687334041393337655454193340, 2.92801790437626255193970967463, 3.29982501017726773349076263126, 3.91218329087799270974986489570, 3.92066980835038460154751971117, 4.79416373267360741953969097080, 4.99055943274169024918634350643, 5.21341032749765676133468647104, 5.60616039871882773883519767843, 5.64795850130428998962826749704, 5.71759443496550371006168742915, 6.25983510352376037865786824071, 6.32901422867944884315413789571, 6.61332342840504900454469836819, 7.03095525451888981842280536482, 7.44521390189786916926397691729, 7.51891451909267780576667476718, 7.982852870821903329330192327278, 8.207055330665520210720896398354, 8.708148978305236828187232442431, 8.805857313211130631884419535493, 9.341850737456151998212655471552
