| L(s) = 1 | − 2·4-s − 2·9-s − 4·13-s + 3·16-s − 2·17-s − 4·25-s + 2·29-s + 4·36-s − 2·37-s − 2·49-s + 8·52-s − 2·53-s − 2·61-s − 4·64-s + 4·68-s + 2·73-s + 3·81-s − 2·89-s + 8·100-s − 2·101-s − 2·109-s + 2·113-s − 4·116-s + 8·117-s − 121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2·4-s − 2·9-s − 4·13-s + 3·16-s − 2·17-s − 4·25-s + 2·29-s + 4·36-s − 2·37-s − 2·49-s + 8·52-s − 2·53-s − 2·61-s − 4·64-s + 4·68-s + 2·73-s + 3·81-s − 2·89-s + 8·100-s − 2·101-s − 2·109-s + 2·113-s − 4·116-s + 8·117-s − 121-s + 127-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 3^{8} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.02594672942\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.02594672942\) |
| \(L(1)\) |
|
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 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 5 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 71 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 73 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 79 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 83 | $C_2$$\times$$C_2^2$ | \( ( 1 + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 89 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
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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
−6.25500001055300190320458265266, −6.16007309175873238074459147244, −5.77676675814051124028833059003, −5.58717946237346780385369085788, −5.51270502230422786508177544075, −5.16571394273538266679996097262, −5.08361688587282165193779316352, −4.90904345705947553797700521156, −4.76900639652470762333503921009, −4.46675937939781558362185920320, −4.26046776857026395185603693903, −4.22183974293025212001512545146, −4.10885032811200299268304876673, −3.38993849866298236774121622613, −3.27329616628742359879406570529, −3.24878350069210109116494865495, −3.17047868209011033298413230621, −2.45528202032532448169567856776, −2.38574991019229485305524845692, −2.35882181545920643829113671474, −2.03457089197386359433099461883, −1.52583904989977116921717146637, −1.45111164042717396505244491998, −0.39868599449444883740415645546, −0.14077357386162249248709802750,
0.14077357386162249248709802750, 0.39868599449444883740415645546, 1.45111164042717396505244491998, 1.52583904989977116921717146637, 2.03457089197386359433099461883, 2.35882181545920643829113671474, 2.38574991019229485305524845692, 2.45528202032532448169567856776, 3.17047868209011033298413230621, 3.24878350069210109116494865495, 3.27329616628742359879406570529, 3.38993849866298236774121622613, 4.10885032811200299268304876673, 4.22183974293025212001512545146, 4.26046776857026395185603693903, 4.46675937939781558362185920320, 4.76900639652470762333503921009, 4.90904345705947553797700521156, 5.08361688587282165193779316352, 5.16571394273538266679996097262, 5.51270502230422786508177544075, 5.58717946237346780385369085788, 5.77676675814051124028833059003, 6.16007309175873238074459147244, 6.25500001055300190320458265266
