| L(s) = 1 | + 2·4-s − 4·7-s + 16-s + 4·19-s + 2·25-s − 8·28-s + 6·49-s − 2·64-s + 8·76-s − 81-s − 4·97-s + 4·100-s + 4·103-s + 4·109-s − 4·112-s + 4·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + ⋯ |
| L(s) = 1 | + 2·4-s − 4·7-s + 16-s + 4·19-s + 2·25-s − 8·28-s + 6·49-s − 2·64-s + 8·76-s − 81-s − 4·97-s + 4·100-s + 4·103-s + 4·109-s − 4·112-s + 4·121-s + 127-s + 131-s − 16·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 4·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 31^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.637198146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.637198146\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2^2$ | \( 1 + T^{4} \) |
| 31 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 5 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 17 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 23 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 29 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 41 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 47 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 71 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 97 | $C_2$ | \( ( 1 + T + T^{2} )^{4} \) |
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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
−6.54422995272007302628291722987, −6.13763933541983825163901961304, −6.13639941030635874873371993620, −5.92650976400184286657049428841, −5.88299681696698616015586660085, −5.43739290253227455999497415915, −5.24853834443573597026867746156, −5.13192353058541479531852847949, −4.74564162473609554576004297098, −4.66127209099612148289433984181, −4.33805947189309395605803513125, −3.90338056539749457643568955549, −3.75134002931906362760479370466, −3.39158470184894353454676696649, −3.24998516280696102666979023987, −3.14017867283196224011842026987, −2.95946830647450877739971147576, −2.92684772818347502238000761039, −2.76575841947642098472929265355, −2.27262546913389382390771439517, −2.03572221756490120652834804377, −1.68411855814244899286713657560, −1.33072446465998332294655024973, −0.898493559848783482792827832102, −0.59442430538662843062705787872,
0.59442430538662843062705787872, 0.898493559848783482792827832102, 1.33072446465998332294655024973, 1.68411855814244899286713657560, 2.03572221756490120652834804377, 2.27262546913389382390771439517, 2.76575841947642098472929265355, 2.92684772818347502238000761039, 2.95946830647450877739971147576, 3.14017867283196224011842026987, 3.24998516280696102666979023987, 3.39158470184894353454676696649, 3.75134002931906362760479370466, 3.90338056539749457643568955549, 4.33805947189309395605803513125, 4.66127209099612148289433984181, 4.74564162473609554576004297098, 5.13192353058541479531852847949, 5.24853834443573597026867746156, 5.43739290253227455999497415915, 5.88299681696698616015586660085, 5.92650976400184286657049428841, 6.13639941030635874873371993620, 6.13763933541983825163901961304, 6.54422995272007302628291722987
