L(s) = 1 | − 6·17-s + 3·25-s − 6·41-s − 12·61-s + 64-s + 6·73-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
L(s) = 1 | − 6·17-s + 3·25-s − 6·41-s − 12·61-s + 64-s + 6·73-s + 6·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + 227-s + 229-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 5^{12} \cdot 37^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.07624344228\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.07624344228\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T^{6} + T^{12} \) |
| 5 | \( ( 1 - T^{2} + T^{4} )^{3} \) |
| 37 | \( 1 - T^{6} + T^{12} \) |
good | 3 | \( 1 - T^{12} + T^{24} \) |
| 7 | \( 1 - T^{12} + T^{24} \) |
| 11 | \( ( 1 - T^{2} + T^{4} )^{6} \) |
| 13 | \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \) |
| 17 | \( ( 1 + T + T^{2} )^{6}( 1 + T^{3} + T^{6} )^{2} \) |
| 19 | \( 1 - T^{12} + T^{24} \) |
| 23 | \( ( 1 - T^{2} + T^{4} )^{6} \) |
| 29 | \( ( 1 - T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \) |
| 31 | \( ( 1 + T^{4} )^{6} \) |
| 41 | \( ( 1 + T + T^{2} )^{6}( 1 - T^{3} + T^{6} )^{2} \) |
| 43 | \( ( 1 + T^{2} )^{12} \) |
| 47 | \( ( 1 - T^{4} + T^{8} )^{3} \) |
| 53 | \( ( 1 + T^{3} + T^{6} )^{2}( 1 - T^{6} + T^{12} ) \) |
| 59 | \( 1 - T^{12} + T^{24} \) |
| 61 | \( ( 1 + T )^{12}( 1 - T^{6} + T^{12} ) \) |
| 67 | \( 1 - T^{12} + T^{24} \) |
| 71 | \( ( 1 - T^{3} + T^{6} )^{2}( 1 + T^{3} + T^{6} )^{2} \) |
| 73 | \( ( 1 - T + T^{2} )^{6}( 1 - T^{2} + T^{4} )^{3} \) |
| 79 | \( 1 - T^{12} + T^{24} \) |
| 83 | \( 1 - T^{12} + T^{24} \) |
| 89 | \( ( 1 + T^{2} )^{6}( 1 + T^{3} + T^{6} )^{2} \) |
| 97 | \( ( 1 - T^{6} + T^{12} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.56630428540586114876625315696, −3.41118207752876735468449731671, −3.38828621985639590162423941445, −3.36184895739625839583905019400, −3.31002131445263096759138357195, −3.27384209442318755550405282664, −3.16738692510039186285834946474, −2.94798617079922184621483765315, −2.93857423065377951281780404827, −2.58921980027080127982068932535, −2.52320938570822896644558510632, −2.46081872740519679359040361226, −2.35959192368561298468016594127, −2.30757495317234191123677241546, −2.20053531237598777928723014866, −2.19702367912220988542991785592, −1.95533852789938805872682765108, −1.89788314147824459590205165907, −1.62788211661275319159173968068, −1.57231095733345363035232617912, −1.36393930898467415438610800549, −1.34764532820558456527781263646, −1.15515603179824714247334384741, −1.08779499773764868223892473897, −0.26425341263058568824718342544,
0.26425341263058568824718342544, 1.08779499773764868223892473897, 1.15515603179824714247334384741, 1.34764532820558456527781263646, 1.36393930898467415438610800549, 1.57231095733345363035232617912, 1.62788211661275319159173968068, 1.89788314147824459590205165907, 1.95533852789938805872682765108, 2.19702367912220988542991785592, 2.20053531237598777928723014866, 2.30757495317234191123677241546, 2.35959192368561298468016594127, 2.46081872740519679359040361226, 2.52320938570822896644558510632, 2.58921980027080127982068932535, 2.93857423065377951281780404827, 2.94798617079922184621483765315, 3.16738692510039186285834946474, 3.27384209442318755550405282664, 3.31002131445263096759138357195, 3.36184895739625839583905019400, 3.38828621985639590162423941445, 3.41118207752876735468449731671, 3.56630428540586114876625315696
Plot not available for L-functions of degree greater than 10.