| L(s) = 1 | − 2·11-s − 2·13-s + 16-s − 2·17-s + 2·23-s + 2·37-s + 4·41-s − 2·47-s − 2·61-s + 4·67-s − 2·71-s − 2·81-s − 4·101-s − 2·103-s + 2·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 4·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯ |
| L(s) = 1 | − 2·11-s − 2·13-s + 16-s − 2·17-s + 2·23-s + 2·37-s + 4·41-s − 2·47-s − 2·61-s + 4·67-s − 2·71-s − 2·81-s − 4·101-s − 2·103-s + 2·113-s + 3·121-s + 127-s + 131-s + 137-s + 139-s + 4·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 2·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 151^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(5^{8} \cdot 151^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2278930632\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2278930632\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 151 | $C_1$ | \( ( 1 + T )^{4} \) |
| good | 2 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 3 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 7 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 17 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 19 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 23 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 29 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 37 | $C_2$$\times$$C_2^2$ | \( ( 1 - T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{4} \) |
| 43 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
| 47 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + T^{2} )^{2}( 1 - T^{2} + T^{4} ) \) |
| 53 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 - T^{2} + T^{4} )^{2} \) |
| 61 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 67 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{4}( 1 + T^{2} )^{2} \) |
| 71 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{4}( 1 - T + T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_2^2$ | \( ( 1 + T^{4} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )^{2}( 1 + T + T^{2} )^{2} \) |
| 97 | $C_2^3$ | \( 1 - T^{4} + T^{8} \) |
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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
−6.26698932377193415201501843967, −5.88001096341790837667408931931, −5.83418588509310746838925390799, −5.67661789250750890511155042293, −5.14855738677265996569117221149, −5.13729129263027524347451432408, −5.07498832077134458607358644133, −5.00237437499491131505455817605, −4.62008518545521506311519675415, −4.35997090487923389782462344712, −4.21397596455068792368686015092, −4.20083347662876051360212144444, −3.74137808789414708611304065603, −3.59841228043469439806557205949, −3.14965956379627893567831643034, −2.93509347421558947516341303610, −2.63183826485035768384900981497, −2.58534993614868273007458775046, −2.56236670861862554444678203713, −2.34782574101151607311151166072, −2.03017696679690753595734893585, −1.35856896675110074181746868083, −1.24567582961438667501522009997, −1.04391565499058485514727675621, −0.17632051744551241665083273977,
0.17632051744551241665083273977, 1.04391565499058485514727675621, 1.24567582961438667501522009997, 1.35856896675110074181746868083, 2.03017696679690753595734893585, 2.34782574101151607311151166072, 2.56236670861862554444678203713, 2.58534993614868273007458775046, 2.63183826485035768384900981497, 2.93509347421558947516341303610, 3.14965956379627893567831643034, 3.59841228043469439806557205949, 3.74137808789414708611304065603, 4.20083347662876051360212144444, 4.21397596455068792368686015092, 4.35997090487923389782462344712, 4.62008518545521506311519675415, 5.00237437499491131505455817605, 5.07498832077134458607358644133, 5.13729129263027524347451432408, 5.14855738677265996569117221149, 5.67661789250750890511155042293, 5.83418588509310746838925390799, 5.88001096341790837667408931931, 6.26698932377193415201501843967
