| L(s) = 1 | − 2·5-s + 6·17-s + 2·25-s + 2·29-s − 2·37-s − 2·41-s − 2·61-s + 2·73-s − 12·85-s + 2·89-s + 2·97-s − 4·109-s + 2·113-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 2·5-s + 6·17-s + 2·25-s + 2·29-s − 2·37-s − 2·41-s − 2·61-s + 2·73-s − 12·85-s + 2·89-s + 2·97-s − 4·109-s + 2·113-s + 127-s + 131-s + 137-s + 139-s − 4·145-s + 149-s + 151-s + 157-s + 163-s + 167-s + 169-s + 173-s + 179-s + 181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{20} \cdot 3^{8} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.503934039\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.503934039\) |
| \(L(1)\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 13 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| good | 5 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 17 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 - T + T^{2} )^{2} \) |
| 19 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T + T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 53 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 59 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 71 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 73 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 97 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
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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
−6.20553771039038958587849891864, −6.18919253940049553504801980958, −5.82270676091732378725884269522, −5.34561533249153893946541770811, −5.26220461683885391746245343073, −5.14193745750741138359260308577, −5.11820843471572399712836447012, −5.09522212394471964634323058201, −4.70469289461819598889127136791, −4.19043893954116615040335475404, −4.07176617135616662461978745322, −3.93907879851800522541227720846, −3.83565207453848391749512822863, −3.32278746460919368887500278619, −3.27167742674920722005692888172, −3.25241960372024887518492999317, −3.04369909215480189073749192157, −2.98486485462025534269904966509, −2.40246568227012780599992510538, −2.12732017817196878923136902101, −1.64856804734113950994667781145, −1.34221916278073917728520722665, −1.32659830745744917726889701514, −0.851595361035763881976507773479, −0.59431611058084717750826382008,
0.59431611058084717750826382008, 0.851595361035763881976507773479, 1.32659830745744917726889701514, 1.34221916278073917728520722665, 1.64856804734113950994667781145, 2.12732017817196878923136902101, 2.40246568227012780599992510538, 2.98486485462025534269904966509, 3.04369909215480189073749192157, 3.25241960372024887518492999317, 3.27167742674920722005692888172, 3.32278746460919368887500278619, 3.83565207453848391749512822863, 3.93907879851800522541227720846, 4.07176617135616662461978745322, 4.19043893954116615040335475404, 4.70469289461819598889127136791, 5.09522212394471964634323058201, 5.11820843471572399712836447012, 5.14193745750741138359260308577, 5.26220461683885391746245343073, 5.34561533249153893946541770811, 5.82270676091732378725884269522, 6.18919253940049553504801980958, 6.20553771039038958587849891864
