| L(s) = 1 | − 4·3-s − 2·4-s + 10·9-s + 8·12-s + 3·16-s + 4·17-s − 20·23-s − 2·25-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 + 8·75-s + 35·81-s − 16·87-s + 40·92-s + 4·100-s − 44·101-s − 16·103-s + 40·108-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 − 2/5·25-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 + 0.923·75-s + 35/9·81-s − 1.71·87-s + 4.17·92-s + 2/5·100-s − 4.37·101-s − 1.57·103-s + 3.84·108-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 5^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4552862758\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4552862758\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 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
−8.061410447807527070752279740101, −7.921734429351138403206818904362, −7.911395839490672226612383495233, −7.28408370850552558868117719050, −7.15770020134942168082494972496, −6.96958288781816199857496169008, −6.75684921745809903690553621907, −6.01134239533421212758117675398, −6.00583179444314314035526684452, −5.88514292279929572146331415696, −5.67092512818433186732684878306, −5.66855613071576167916315318375, −5.28244379914479056395443846226, −4.67881252431924868672926182178, −4.44528630915969865537365404958, −4.36571903843325057029973807942, −4.02711500343632160153733806308, −3.96000082420657021028174005107, −3.49828169818915795469979154788, −2.95706065298897152912152530358, −2.30692486753724618079884139079, −2.16506396219071637825830815672, −1.47051304109435739982787953074, −0.913082115461151326886198426371, −0.42435673838783386560341274055,
0.42435673838783386560341274055, 0.913082115461151326886198426371, 1.47051304109435739982787953074, 2.16506396219071637825830815672, 2.30692486753724618079884139079, 2.95706065298897152912152530358, 3.49828169818915795469979154788, 3.96000082420657021028174005107, 4.02711500343632160153733806308, 4.36571903843325057029973807942, 4.44528630915969865537365404958, 4.67881252431924868672926182178, 5.28244379914479056395443846226, 5.66855613071576167916315318375, 5.67092512818433186732684878306, 5.88514292279929572146331415696, 6.00583179444314314035526684452, 6.01134239533421212758117675398, 6.75684921745809903690553621907, 6.96958288781816199857496169008, 7.15770020134942168082494972496, 7.28408370850552558868117719050, 7.911395839490672226612383495233, 7.921734429351138403206818904362, 8.061410447807527070752279740101
