| L(s) = 1 | − 4·25-s − 64·43-s − 44·49-s + 8·61-s + 32·67-s + 128·103-s − 32·109-s − 76·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 92·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + 197-s + 199-s + 211-s + 223-s + ⋯ |
| L(s) = 1 | − 4/5·25-s − 9.75·43-s − 6.28·49-s + 1.02·61-s + 3.90·67-s + 12.6·103-s − 3.06·109-s − 6.90·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 7.07·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + 0.0712·197-s + 0.0708·199-s + 0.0688·211-s + 0.0669·223-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{72} \cdot 3^{24} \cdot 5^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3639954686\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3639954686\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + 4 T^{2} - 17 T^{4} - 152 T^{6} - 17 p^{2} T^{8} + 4 p^{4} T^{10} + p^{6} T^{12} \) |
| good | 7 | \( ( 1 + 11 T^{2} - 8 T^{3} + 11 p T^{4} + p^{3} T^{6} )^{4} \) |
| 11 | \( ( 1 + 38 T^{2} + 811 T^{4} + 10796 T^{6} + 811 p^{2} T^{8} + 38 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 13 | \( ( 1 - 46 T^{2} + 71 p T^{4} - 12828 T^{6} + 71 p^{3} T^{8} - 46 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 17 | \( ( 1 + 4 p T^{2} + 2311 T^{4} + 48968 T^{6} + 2311 p^{2} T^{8} + 4 p^{5} T^{10} + p^{6} T^{12} )^{2} \) |
| 19 | \( ( 1 - 34 T^{2} + 935 T^{4} - 20604 T^{6} + 935 p^{2} T^{8} - 34 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 23 | \( ( 1 - 50 T^{2} + 367 T^{4} + 12196 T^{6} + 367 p^{2} T^{8} - 50 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 29 | \( ( 1 - 76 T^{2} + 3391 T^{4} - 114424 T^{6} + 3391 p^{2} T^{8} - 76 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 31 | \( ( 1 - 98 T^{2} + 4927 T^{4} - 170300 T^{6} + 4927 p^{2} T^{8} - 98 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 37 | \( ( 1 - 178 T^{2} + 14155 T^{4} - 661348 T^{6} + 14155 p^{2} T^{8} - 178 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 41 | \( ( 1 - 144 T^{2} + 11315 T^{4} - 13856 p T^{6} + 11315 p^{2} T^{8} - 144 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 43 | \( ( 1 + 16 T + 137 T^{2} + 864 T^{3} + 137 p T^{4} + 16 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 47 | \( ( 1 - 106 T^{2} + 5743 T^{4} - 227980 T^{6} + 5743 p^{2} T^{8} - 106 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 53 | \( ( 1 + 12 T^{2} + 7727 T^{4} + 58168 T^{6} + 7727 p^{2} T^{8} + 12 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 59 | \( ( 1 + 134 T^{2} + 15947 T^{4} + 1000044 T^{6} + 15947 p^{2} T^{8} + 134 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 61 | \( ( 1 - 2 T + 83 T^{2} + 84 T^{3} + 83 p T^{4} - 2 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 67 | \( ( 1 - 8 T + 121 T^{2} - 560 T^{3} + 121 p T^{4} - 8 p^{2} T^{5} + p^{3} T^{6} )^{4} \) |
| 71 | \( ( 1 + 170 T^{2} + 18527 T^{4} + 1427916 T^{6} + 18527 p^{2} T^{8} + 170 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 73 | \( ( 1 - 162 T^{2} + 15215 T^{4} - 1057532 T^{6} + 15215 p^{2} T^{8} - 162 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 79 | \( ( 1 - 130 T^{2} + 14495 T^{4} - 1438332 T^{6} + 14495 p^{2} T^{8} - 130 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 83 | \( ( 1 - 242 T^{2} + 33959 T^{4} - 3241692 T^{6} + 33959 p^{2} T^{8} - 242 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 89 | \( ( 1 - 416 T^{2} + 78163 T^{4} - 8732480 T^{6} + 78163 p^{2} T^{8} - 416 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
| 97 | \( ( 1 - 306 T^{2} + 58751 T^{4} - 6745628 T^{6} + 58751 p^{2} T^{8} - 306 p^{4} T^{10} + p^{6} T^{12} )^{2} \) |
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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
−2.67381126613234837659134202898, −2.57789106119193146461184398947, −2.54806836454655653355020234768, −2.31885963812657792555625357936, −2.27906855582422455653429859892, −2.21701204833608061034442761110, −2.17678543536411096798628560568, −2.14277046792175239250674500509, −1.80771973759042667520257007468, −1.74546970085067433834283195384, −1.70855071497027814531259773186, −1.62964820557094025374646080641, −1.57783681797168389372532059095, −1.56774433377414810135813291705, −1.52486725184267553296531639877, −1.43614823532505035143775436601, −1.22433243025777380170459710937, −1.13135085601375212226271208391, −0.823946803602608484857880755478, −0.812506308284906181386012618162, −0.63303155748892544149562888380, −0.55364345970415237778214291617, −0.28249644840620816916953760531, −0.11620181057265152776288458495, −0.10652193752622463819029293997,
0.10652193752622463819029293997, 0.11620181057265152776288458495, 0.28249644840620816916953760531, 0.55364345970415237778214291617, 0.63303155748892544149562888380, 0.812506308284906181386012618162, 0.823946803602608484857880755478, 1.13135085601375212226271208391, 1.22433243025777380170459710937, 1.43614823532505035143775436601, 1.52486725184267553296531639877, 1.56774433377414810135813291705, 1.57783681797168389372532059095, 1.62964820557094025374646080641, 1.70855071497027814531259773186, 1.74546970085067433834283195384, 1.80771973759042667520257007468, 2.14277046792175239250674500509, 2.17678543536411096798628560568, 2.21701204833608061034442761110, 2.27906855582422455653429859892, 2.31885963812657792555625357936, 2.54806836454655653355020234768, 2.57789106119193146461184398947, 2.67381126613234837659134202898
Plot not available for L-functions of degree greater than 10.
