| L(s) = 1 | + 8·3-s − 32·9-s − 56·11-s + 128·17-s + 408·19-s + 12·25-s − 488·27-s − 448·33-s − 984·41-s + 856·43-s − 348·49-s + 1.02e3·51-s + 3.26e3·57-s + 2.45e3·59-s + 1.30e3·67-s − 1.66e3·73-s + 96·75-s − 466·81-s + 1.41e3·83-s − 128·89-s − 384·97-s + 1.79e3·99-s + 136·107-s + 1.08e3·113-s − 3.36e3·121-s − 7.87e3·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.53·3-s − 1.18·9-s − 1.53·11-s + 1.82·17-s + 4.92·19-s + 0.0959·25-s − 3.47·27-s − 2.36·33-s − 3.74·41-s + 3.03·43-s − 1.01·49-s + 2.81·51-s + 7.58·57-s + 5.41·59-s + 2.37·67-s − 2.66·73-s + 0.147·75-s − 0.639·81-s + 1.87·83-s − 0.152·89-s − 0.401·97-s + 1.81·99-s + 0.122·107-s + 0.899·113-s − 2.52·121-s − 5.77·123-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(4-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{36}\right)^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.836218516\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.836218516\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( ( 1 - 4 T + 40 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 5 | $C_2^2:C_4$ | \( 1 - 12 T^{2} - 1482 T^{4} - 12 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2:C_4$ | \( 1 + 348 T^{2} + 134502 T^{4} + 348 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 28 T + 2856 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 84 T^{2} + 8049750 T^{4} + 84 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 64 T + 10050 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 19 | $D_{4}$ | \( ( 1 - 204 T + 23784 T^{2} - 204 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 23 | $C_2^2:C_4$ | \( 1 + 31260 T^{2} + 471031590 T^{4} + 31260 p^{6} T^{6} + p^{12} T^{8} \) |
| 29 | $C_2^2:C_4$ | \( 1 + 60180 T^{2} + 1747419030 T^{4} + 60180 p^{6} T^{6} + p^{12} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 32388 T^{2} + 927859590 T^{4} - 32388 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 152948 T^{2} + 10364702902 T^{4} + 152948 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 + 12 p T + 165590 T^{2} + 12 p^{4} T^{3} + p^{6} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 - 428 T + 182760 T^{2} - 428 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 333372 T^{2} + 47698486086 T^{4} + 333372 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 4580 p T^{2} + 48450398710 T^{4} + 4580 p^{7} T^{6} + p^{12} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 - 1228 T + 746856 T^{2} - 1228 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 133268 T^{2} + 27019375126 T^{4} + 133268 p^{6} T^{6} + p^{12} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 652 T + 488680 T^{2} - 652 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 + 300124 T^{2} + 278020688998 T^{4} + 300124 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 832 T + 948498 T^{2} + 832 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 79 | $C_2^2:C_4$ | \( 1 + 886716 T^{2} + 560936541894 T^{4} + 886716 p^{6} T^{6} + p^{12} T^{8} \) |
| 83 | $D_{4}$ | \( ( 1 - 708 T + 1254440 T^{2} - 708 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 64 T + 1327730 T^{2} + 64 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 97 | $D_{4}$ | \( ( 1 + 192 T + 1318434 T^{2} + 192 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
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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
−7.59377362688224010949512807121, −7.26799497593254846929135233131, −7.06119110361927215049699102967, −6.89731028084663087390825079132, −6.65890194943539375458980351798, −5.90714184065969115314463298131, −5.82344014660970069025055426623, −5.56774456341761764245497772318, −5.51300775727922335223657553601, −5.19610136202355050664927567208, −5.12605603451279643053630863085, −4.90146890982607722813999855244, −4.31197186529396437303376502765, −3.70721337643650104638109893336, −3.49346166764847817404970322583, −3.48272480437994026252146216515, −3.23073467808983280807178901160, −2.89919675522409391045441324575, −2.69232896957436845444102682344, −2.39092458351548993194242301191, −2.16087050611377193273132058235, −1.45298036519301538271147161365, −1.04812047972405955696001018415, −0.807086783635096112862098041001, −0.34578031227220694063316930445,
0.34578031227220694063316930445, 0.807086783635096112862098041001, 1.04812047972405955696001018415, 1.45298036519301538271147161365, 2.16087050611377193273132058235, 2.39092458351548993194242301191, 2.69232896957436845444102682344, 2.89919675522409391045441324575, 3.23073467808983280807178901160, 3.48272480437994026252146216515, 3.49346166764847817404970322583, 3.70721337643650104638109893336, 4.31197186529396437303376502765, 4.90146890982607722813999855244, 5.12605603451279643053630863085, 5.19610136202355050664927567208, 5.51300775727922335223657553601, 5.56774456341761764245497772318, 5.82344014660970069025055426623, 5.90714184065969115314463298131, 6.65890194943539375458980351798, 6.89731028084663087390825079132, 7.06119110361927215049699102967, 7.26799497593254846929135233131, 7.59377362688224010949512807121
