| L(s) = 1 | − 6·4-s − 4·7-s + 21·16-s + 24·28-s − 8·37-s + 12·43-s + 22·49-s − 56·64-s − 64·67-s + 56·79-s − 4·81-s − 56·109-s − 84·112-s + 40·121-s + 127-s + 131-s + 137-s + 139-s + 48·148-s + 149-s + 151-s + 157-s + 163-s + 167-s + 30·169-s − 72·172-s + 173-s + ⋯ |
| L(s) = 1 | − 3·4-s − 1.51·7-s + 21/4·16-s + 4.53·28-s − 1.31·37-s + 1.82·43-s + 22/7·49-s − 7·64-s − 7.81·67-s + 6.30·79-s − 4/9·81-s − 5.36·109-s − 7.93·112-s + 3.63·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 3.94·148-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 2.30·169-s − 5.48·172-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{12} \cdot 5^{24} \cdot 7^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9895069932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9895069932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( ( 1 + T^{2} )^{6} \) |
| 3 | \( 1 + 4 T^{4} - 10 p T^{6} + 4 p^{2} T^{8} + p^{6} T^{12} \) |
| 5 | \( 1 \) |
| 7 | \( ( 1 + 2 T - 5 T^{2} - 36 T^{3} - 5 p T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{2} \) |
| good | 11 | \( ( 1 - 20 T^{2} + 320 T^{4} - 3962 T^{6} + 320 p^{2} T^{8} - 20 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 15 T^{2} + 43 p T^{4} - 5110 T^{6} + 43 p^{3} T^{8} - 15 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 + 29 T^{2} + 494 T^{4} + 5297 T^{6} + 494 p^{2} T^{8} + 29 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 96 T^{2} + 4132 T^{4} - 101374 T^{6} + 4132 p^{2} T^{8} - 96 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 73 T^{2} + 105 p T^{4} - 58154 T^{6} + 105 p^{3} T^{8} - 73 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 109 T^{2} + 5535 T^{4} - 186434 T^{6} + 5535 p^{2} T^{8} - 109 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 51 T^{2} + 2623 T^{4} - 73486 T^{6} + 2623 p^{2} T^{8} - 51 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 + 2 T + p T^{2} + 300 T^{3} + p^{2} T^{4} + 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 41 | \( ( 1 - 3 T^{2} + 3166 T^{4} - 23 p T^{6} + 3166 p^{2} T^{8} - 3 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 - 3 T + 109 T^{2} - 266 T^{3} + 109 p T^{4} - 3 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 47 | \( ( 1 + 78 T^{2} + 7315 T^{4} + 322124 T^{6} + 7315 p^{2} T^{8} + 78 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 - 165 T^{2} + 13615 T^{4} - 813650 T^{6} + 13615 p^{2} T^{8} - 165 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 + 179 T^{2} + 18587 T^{4} + 1260626 T^{6} + 18587 p^{2} T^{8} + 179 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 3 p T^{2} + 12451 T^{4} - 609898 T^{6} + 12451 p^{2} T^{8} - 3 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 67 | \( ( 1 + 16 T + 258 T^{2} + 2108 T^{3} + 258 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 71 | \( ( 1 - 210 T^{2} + 21667 T^{4} - 1617716 T^{6} + 21667 p^{2} T^{8} - 210 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 332 T^{2} + 52356 T^{4} - 4849846 T^{6} + 52356 p^{2} T^{8} - 332 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 - 14 T + 193 T^{2} - 2172 T^{3} + 193 p T^{4} - 14 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 83 | \( ( 1 + 309 T^{2} + 44338 T^{4} + 4238477 T^{6} + 44338 p^{2} T^{8} + 309 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 + 348 T^{2} + 58036 T^{4} + 6184130 T^{6} + 58036 p^{2} T^{8} + 348 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 - 158 T^{2} + 21375 T^{4} - 2986564 T^{6} + 21375 p^{2} T^{8} - 158 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
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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
−3.30930975316153323217875783456, −3.17985578472146277016954228934, −2.89233189741446259823161333007, −2.88436279169858277173064115710, −2.81099746480852087851221595631, −2.69306053195180120310704117757, −2.61841997007074116825203943452, −2.55278126988216747032770406450, −2.34800545086688919419120226124, −2.32184513858266542252848332757, −2.29685301149355771755722748684, −2.06829389242040314291249005723, −1.86885243236405861684298872184, −1.69784905881433644529121566763, −1.69507514327082191990121727703, −1.44164373011736851716524790250, −1.41374266327413573039296007045, −1.38924976510596922884669902560, −1.10115525474682899880370079157, −0.924145639069131950334617819101, −0.847913853050507608860301150241, −0.64688343361717860049380479484, −0.40513456782565977058840835953, −0.31753769077783261438014609348, −0.20222678518793418210958281516,
0.20222678518793418210958281516, 0.31753769077783261438014609348, 0.40513456782565977058840835953, 0.64688343361717860049380479484, 0.847913853050507608860301150241, 0.924145639069131950334617819101, 1.10115525474682899880370079157, 1.38924976510596922884669902560, 1.41374266327413573039296007045, 1.44164373011736851716524790250, 1.69507514327082191990121727703, 1.69784905881433644529121566763, 1.86885243236405861684298872184, 2.06829389242040314291249005723, 2.29685301149355771755722748684, 2.32184513858266542252848332757, 2.34800545086688919419120226124, 2.55278126988216747032770406450, 2.61841997007074116825203943452, 2.69306053195180120310704117757, 2.81099746480852087851221595631, 2.88436279169858277173064115710, 2.89233189741446259823161333007, 3.17985578472146277016954228934, 3.30930975316153323217875783456
Plot not available for L-functions of degree greater than 10.
