| L(s) = 1 | − 2·3-s − 12·7-s + 2·9-s + 24·21-s + 10·25-s − 6·27-s + 24·29-s + 72·49-s − 24·63-s − 20·75-s + 11·81-s + 44·83-s − 48·87-s + 72·101-s + 36·103-s − 52·107-s + 44·121-s + 127-s + 131-s + 137-s + 139-s − 144·147-s + 149-s + 151-s + 157-s + 163-s + 167-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 4.53·7-s + 2/3·9-s + 5.23·21-s + 2·25-s − 1.15·27-s + 4.45·29-s + 72/7·49-s − 3.02·63-s − 2.30·75-s + 11/9·81-s + 4.82·83-s − 5.14·87-s + 7.16·101-s + 3.54·103-s − 5.02·107-s + 4·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s − 11.8·147-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{28} \cdot 3^{4} \cdot 5^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9718538066\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9718538066\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 7 | $C_2^2$ | \( ( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 23 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 + 62 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} )( 1 + 18 T + 162 T^{2} + 18 p T^{3} + p^{2} T^{4} ) \) |
| 47 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 14 T + 98 T^{2} - 14 p T^{3} + p^{2} T^{4} )( 1 + 14 T + 98 T^{2} + 14 p T^{3} + p^{2} T^{4} ) \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 59 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 61 | $C_2^2$ | \( ( 1 - 58 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$$\times$$C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} ) \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{4} \) |
| 73 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 - 22 T + 242 T^{2} - 22 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2}( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{4} \) |
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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
−6.41107655812277498565487239646, −6.36251399546116178535706637469, −6.14661796493126773161628836506, −6.10873500084234415115075507025, −6.09605387708813105993272224185, −5.43591226437249346217464617614, −5.35084272240477023762877683464, −5.08594753628017057861796356629, −4.82499018857933670360990128283, −4.64221672376206259379062627607, −4.50169762663541716834269026783, −4.09567646597640512895996247776, −3.91808875249130013207707724221, −3.34431769883903162803898894090, −3.30025837259237129324122759116, −3.25812768095409162703779132147, −3.23747753694142801973158741297, −2.74350528394313331854383477052, −2.52506338656078348156080828548, −2.08059419352610335641479018500, −2.07863211689048382151470132373, −1.07180392675020249819394692854, −0.830386226316714201500292802031, −0.70435770903023683861291531643, −0.33406559510895863014558709123,
0.33406559510895863014558709123, 0.70435770903023683861291531643, 0.830386226316714201500292802031, 1.07180392675020249819394692854, 2.07863211689048382151470132373, 2.08059419352610335641479018500, 2.52506338656078348156080828548, 2.74350528394313331854383477052, 3.23747753694142801973158741297, 3.25812768095409162703779132147, 3.30025837259237129324122759116, 3.34431769883903162803898894090, 3.91808875249130013207707724221, 4.09567646597640512895996247776, 4.50169762663541716834269026783, 4.64221672376206259379062627607, 4.82499018857933670360990128283, 5.08594753628017057861796356629, 5.35084272240477023762877683464, 5.43591226437249346217464617614, 6.09605387708813105993272224185, 6.10873500084234415115075507025, 6.14661796493126773161628836506, 6.36251399546116178535706637469, 6.41107655812277498565487239646
