| L(s) = 1 | + 4·3-s − 2·4-s + 10·9-s − 8·12-s + 3·16-s − 4·17-s + 20·23-s + 20·27-s + 4·29-s − 20·36-s − 32·43-s + 12·48-s − 16·51-s − 24·53-s − 16·61-s − 4·64-s + 8·68-s + 80·69-s + 35·81-s + 16·87-s − 40·92-s − 44·101-s + 16·103-s − 40·108-s + 12·113-s − 8·116-s + 44·121-s + ⋯ |
| L(s) = 1 | + 2.30·3-s − 4-s + 10/3·9-s − 2.30·12-s + 3/4·16-s − 0.970·17-s + 4.17·23-s + 3.84·27-s + 0.742·29-s − 3.33·36-s − 4.87·43-s + 1.73·48-s − 2.24·51-s − 3.29·53-s − 2.04·61-s − 1/2·64-s + 0.970·68-s + 9.63·69-s + 35/9·81-s + 1.71·87-s − 4.17·92-s − 4.37·101-s + 1.57·103-s − 3.84·108-s + 1.12·113-s − 0.742·116-s + 4·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.900510135\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.900510135\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{4} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 7 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 17 | $D_{4}$ | \( ( 1 + 2 T + 22 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 48 T^{2} + 1246 T^{4} - 48 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 10 T + 58 T^{2} - 10 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 2 T + 46 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 26 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 36 T^{2} + 2230 T^{4} - 36 p^{2} T^{6} + p^{4} T^{8} \) |
| 41 | $D_4\times C_2$ | \( 1 - 28 T^{2} + 230 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{4} \) |
| 47 | $D_4\times C_2$ | \( 1 - 76 T^{2} + 5030 T^{4} - 76 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 9974 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 8 T + 86 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 132 T^{2} + 10006 T^{4} - 132 p^{2} T^{6} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 - 172 T^{2} + 16646 T^{4} - 172 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 216 T^{2} + 21022 T^{4} - 216 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 - 50 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 28 T^{2} - 6826 T^{4} - 28 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 - 124 T^{2} + 6374 T^{4} - 124 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $D_4\times C_2$ | \( 1 - 312 T^{2} + 41854 T^{4} - 312 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.84312221597269205843797275719, −6.29011593065188401812238925527, −6.22420389064173470069771826794, −5.95346426375846843507057266931, −5.63489092925202566628667700759, −5.42604272272907554319016985050, −4.95922689061128231617247030263, −4.70754746366557470192239716234, −4.69797069752167179308593187933, −4.67349646083220932559689189366, −4.65891843262147813886325007731, −4.20843723232626618347097079893, −3.62652705167760180808300020464, −3.48752189985253012042042503852, −3.32477372344949089044051845598, −3.29193333186213728188301094203, −3.00251616478735544647298974404, −2.70613641588438421311108869219, −2.69830541979073193496203521080, −1.87600191227419803787448943768, −1.87123463342331502262106039997, −1.67856806084251347399750267055, −1.25453944220236276667067501143, −0.891295285628353587483195737751, −0.34965443300621170534743459332,
0.34965443300621170534743459332, 0.891295285628353587483195737751, 1.25453944220236276667067501143, 1.67856806084251347399750267055, 1.87123463342331502262106039997, 1.87600191227419803787448943768, 2.69830541979073193496203521080, 2.70613641588438421311108869219, 3.00251616478735544647298974404, 3.29193333186213728188301094203, 3.32477372344949089044051845598, 3.48752189985253012042042503852, 3.62652705167760180808300020464, 4.20843723232626618347097079893, 4.65891843262147813886325007731, 4.67349646083220932559689189366, 4.69797069752167179308593187933, 4.70754746366557470192239716234, 4.95922689061128231617247030263, 5.42604272272907554319016985050, 5.63489092925202566628667700759, 5.95346426375846843507057266931, 6.22420389064173470069771826794, 6.29011593065188401812238925527, 6.84312221597269205843797275719
