| L(s) = 1 | − 2-s + 2·5-s − 4·9-s − 2·10-s + 4·18-s + 25-s + 32-s − 3·37-s − 41-s − 8·45-s + 49-s − 50-s − 2·61-s − 64-s + 2·73-s + 3·74-s + 10·81-s + 82-s + 8·90-s − 5·97-s − 98-s + 5·101-s − 3·113-s + 121-s + 2·122-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 2-s + 2·5-s − 4·9-s − 2·10-s + 4·18-s + 25-s + 32-s − 3·37-s − 41-s − 8·45-s + 49-s − 50-s − 2·61-s − 64-s + 2·73-s + 3·74-s + 10·81-s + 82-s + 8·90-s − 5·97-s − 98-s + 5·101-s − 3·113-s + 121-s + 2·122-s + 127-s + 131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{8} \cdot 41^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1155394926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1155394926\) |
| \(L(1)\) |
|
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 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 41 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| good | 3 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 5 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 7 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 11 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 13 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 17 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 19 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 23 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 29 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 31 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 37 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 - T + T^{2} - T^{3} + T^{4} ) \) |
| 43 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 47 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 53 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 59 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 61 | $C_4$ | \( ( 1 + T + T^{2} + T^{3} + T^{4} )^{2} \) |
| 67 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 71 | $C_4\times C_2$ | \( 1 - T^{2} + T^{4} - T^{6} + T^{8} \) |
| 73 | $C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + T^{2} )^{4} \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{4}( 1 + T )^{4} \) |
| 89 | $C_4$$\times$$C_4$ | \( ( 1 - T + T^{2} - T^{3} + T^{4} )( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
| 97 | $C_1$$\times$$C_4$ | \( ( 1 + T )^{4}( 1 + T + T^{2} + T^{3} + T^{4} ) \) |
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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
−9.632719146874707793402677145655, −9.144277440944001181458604538181, −9.108826497477132658211277265415, −9.101205702288621514215501238496, −8.711166280378404405933828040312, −8.488002706487949380527181328802, −8.095820178973720601315976554995, −7.965951516206819742473072934641, −7.932204997056532795828957939533, −6.93886035143021211153624870729, −6.89944620426769172352808888648, −6.65646052297742784264411778986, −6.12510857055227658078927412167, −5.86407998785445718987111686513, −5.71967667474643560245412925019, −5.60628249265597878547519197620, −5.23869840015675314642057463457, −4.88951706528056482401978081920, −4.53447640657719775660999853148, −3.57109128524881941414714243813, −3.44006597845181513551764232923, −3.03848108752394923544774568067, −2.51903332244233312931291208173, −2.26742282274066328026911668242, −1.73550282431264646745125129550,
1.73550282431264646745125129550, 2.26742282274066328026911668242, 2.51903332244233312931291208173, 3.03848108752394923544774568067, 3.44006597845181513551764232923, 3.57109128524881941414714243813, 4.53447640657719775660999853148, 4.88951706528056482401978081920, 5.23869840015675314642057463457, 5.60628249265597878547519197620, 5.71967667474643560245412925019, 5.86407998785445718987111686513, 6.12510857055227658078927412167, 6.65646052297742784264411778986, 6.89944620426769172352808888648, 6.93886035143021211153624870729, 7.932204997056532795828957939533, 7.965951516206819742473072934641, 8.095820178973720601315976554995, 8.488002706487949380527181328802, 8.711166280378404405933828040312, 9.101205702288621514215501238496, 9.108826497477132658211277265415, 9.144277440944001181458604538181, 9.632719146874707793402677145655
