| L(s) = 1 | + 2·5-s + 2·7-s − 26·11-s + 50·17-s + 10·19-s + 28·23-s − 59·25-s + 4·35-s + 210·43-s + 22·47-s − 163·49-s − 52·55-s + 214·61-s + 102·73-s − 52·77-s − 404·83-s + 100·85-s + 20·95-s + 164·101-s + 56·115-s + 100·119-s − 203·121-s − 34·125-s + 127-s + 131-s + 20·133-s + 137-s + ⋯ |
| L(s) = 1 | + 2/5·5-s + 2/7·7-s − 2.36·11-s + 2.94·17-s + 0.526·19-s + 1.21·23-s − 2.35·25-s + 4/35·35-s + 4.88·43-s + 0.468·47-s − 3.32·49-s − 0.945·55-s + 3.50·61-s + 1.39·73-s − 0.675·77-s − 4.86·83-s + 1.17·85-s + 4/19·95-s + 1.62·101-s + 0.486·115-s + 0.840·119-s − 1.67·121-s − 0.271·125-s + 0.00787·127-s + 0.00763·131-s + 0.150·133-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{24} \cdot 3^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7838636410\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7838636410\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 - 10 T + 407 T^{2} - 388 p T^{3} + 407 p^{2} T^{4} - 10 p^{4} T^{5} + p^{6} T^{6} \) |
| good | 5 | \( ( 1 - T + 31 T^{2} - 74 T^{3} + 31 p^{2} T^{4} - p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 7 | \( ( 1 - T + 83 T^{2} - 262 T^{3} + 83 p^{2} T^{4} - p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 11 | \( ( 1 + 13 T + 355 T^{2} + 3134 T^{3} + 355 p^{2} T^{4} + 13 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 13 | \( 1 - 222 T^{2} - 8913 T^{4} + 8535292 T^{6} - 8913 p^{4} T^{8} - 222 p^{8} T^{10} + p^{12} T^{12} \) |
| 17 | \( ( 1 - 25 T + 835 T^{2} - 14342 T^{3} + 835 p^{2} T^{4} - 25 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 23 | \( ( 1 - 14 T + 1027 T^{2} - 9676 T^{3} + 1027 p^{2} T^{4} - 14 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 29 | \( 1 - 1878 T^{2} + 1963455 T^{4} - 1513144244 T^{6} + 1963455 p^{4} T^{8} - 1878 p^{8} T^{10} + p^{12} T^{12} \) |
| 31 | \( 1 - 2574 T^{2} + 2592063 T^{4} - 2002604132 T^{6} + 2592063 p^{4} T^{8} - 2574 p^{8} T^{10} + p^{12} T^{12} \) |
| 37 | \( 1 - 2238 T^{2} + 4271055 T^{4} - 7604589764 T^{6} + 4271055 p^{4} T^{8} - 2238 p^{8} T^{10} + p^{12} T^{12} \) |
| 41 | \( 1 + 810 T^{2} + 4444335 T^{4} + 84590476 T^{6} + 4444335 p^{4} T^{8} + 810 p^{8} T^{10} + p^{12} T^{12} \) |
| 43 | \( ( 1 - 105 T + 8979 T^{2} - 424054 T^{3} + 8979 p^{2} T^{4} - 105 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 47 | \( ( 1 - 11 T + 4543 T^{2} - 23326 T^{3} + 4543 p^{2} T^{4} - 11 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 53 | \( 1 - 12582 T^{2} + 74706975 T^{4} - 265412742932 T^{6} + 74706975 p^{4} T^{8} - 12582 p^{8} T^{10} + p^{12} T^{12} \) |
| 59 | \( 1 - 1350 T^{2} + 25160031 T^{4} - 18935070932 T^{6} + 25160031 p^{4} T^{8} - 1350 p^{8} T^{10} + p^{12} T^{12} \) |
| 61 | \( ( 1 - 107 T + 11099 T^{2} - 692018 T^{3} + 11099 p^{2} T^{4} - 107 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 67 | \( 1 - 22614 T^{2} + 229583679 T^{4} - 1330212756404 T^{6} + 229583679 p^{4} T^{8} - 22614 p^{8} T^{10} + p^{12} T^{12} \) |
| 71 | \( 1 - 18774 T^{2} + 191507823 T^{4} - 1184211826292 T^{6} + 191507823 p^{4} T^{8} - 18774 p^{8} T^{10} + p^{12} T^{12} \) |
| 73 | \( ( 1 - 51 T + 8091 T^{2} - 428066 T^{3} + 8091 p^{2} T^{4} - 51 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 79 | \( 1 - 17598 T^{2} + 157161279 T^{4} - 1016250978692 T^{6} + 157161279 p^{4} T^{8} - 17598 p^{8} T^{10} + p^{12} T^{12} \) |
| 83 | \( ( 1 + 202 T + 33559 T^{2} + 3043148 T^{3} + 33559 p^{2} T^{4} + 202 p^{4} T^{5} + p^{6} T^{6} )^{2} \) |
| 89 | \( 1 - 44118 T^{2} + 836433135 T^{4} - 8707655419508 T^{6} + 836433135 p^{4} T^{8} - 44118 p^{8} T^{10} + p^{12} T^{12} \) |
| 97 | \( 1 - 48054 T^{2} + 1028309583 T^{4} - 12508133393972 T^{6} + 1028309583 p^{4} T^{8} - 48054 p^{8} T^{10} + p^{12} T^{12} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−4.25007238098037271984229123670, −4.23651206214991550639468485994, −4.13375362361393340208140290237, −3.94922585746564822874755371547, −3.94281481325164339636705683254, −3.61126994928244494636122501296, −3.29520830092816607162755325730, −3.29367877260511241730055276797, −3.08547790025753216452016406608, −3.04293124755836111716094102146, −2.84647678530657696385511381420, −2.80678129845427368167642892147, −2.49732912416342910575139579279, −2.33716589467507955706415446447, −2.19670165142754109191733759039, −2.00035995485280845345911688336, −1.87284435911529408119054900398, −1.71463517325973325382029507665, −1.38332283145928466397597991945, −1.05755291209015067371543240391, −1.05209993051750320186607871022, −0.830166879781777634386541149546, −0.789224482242661836178583186361, −0.32115557435740177551521484377, −0.07135233406288597250521606954,
0.07135233406288597250521606954, 0.32115557435740177551521484377, 0.789224482242661836178583186361, 0.830166879781777634386541149546, 1.05209993051750320186607871022, 1.05755291209015067371543240391, 1.38332283145928466397597991945, 1.71463517325973325382029507665, 1.87284435911529408119054900398, 2.00035995485280845345911688336, 2.19670165142754109191733759039, 2.33716589467507955706415446447, 2.49732912416342910575139579279, 2.80678129845427368167642892147, 2.84647678530657696385511381420, 3.04293124755836111716094102146, 3.08547790025753216452016406608, 3.29367877260511241730055276797, 3.29520830092816607162755325730, 3.61126994928244494636122501296, 3.94281481325164339636705683254, 3.94922585746564822874755371547, 4.13375362361393340208140290237, 4.23651206214991550639468485994, 4.25007238098037271984229123670
Plot not available for L-functions of degree greater than 10.
