| L(s) = 1 | − 4·4-s − 4·13-s + 12·16-s + 100·19-s + 24·31-s − 144·37-s + 116·43-s + 14·49-s + 16·52-s − 80·61-s − 32·64-s + 216·67-s − 300·73-s − 400·76-s − 400·79-s + 72·97-s − 84·103-s + 224·109-s + 252·121-s − 96·124-s + 127-s + 131-s + 137-s + 139-s + 576·148-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s − 0.307·13-s + 3/4·16-s + 5.26·19-s + 0.774·31-s − 3.89·37-s + 2.69·43-s + 2/7·49-s + 4/13·52-s − 1.31·61-s − 1/2·64-s + 3.22·67-s − 4.10·73-s − 5.26·76-s − 5.06·79-s + 0.742·97-s − 0.815·103-s + 2.05·109-s + 2.08·121-s − 0.774·124-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 3.89·148-s + 0.00671·149-s + 0.00662·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(3-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.806344477\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.806344477\) |
| \(L(2)\) |
|
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 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| good | 11 | $D_4\times C_2$ | \( 1 - 252 T^{2} + 34511 T^{4} - 252 p^{4} T^{6} + p^{8} T^{8} \) |
| 13 | $D_{4}$ | \( ( 1 + 2 T + 164 T^{2} + 2 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 17 | $D_4\times C_2$ | \( 1 - 888 T^{2} + 360146 T^{4} - 888 p^{4} T^{6} + p^{8} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 - 50 T + 1340 T^{2} - 50 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 812 T^{2} + 581375 T^{4} - 812 p^{4} T^{6} + p^{8} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 - 2788 T^{2} + 3352695 T^{4} - 2788 p^{4} T^{6} + p^{8} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 - 12 T + 1930 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 37 | $D_{4}$ | \( ( 1 + 72 T + 3187 T^{2} + 72 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 6276 T^{2} + 15449174 T^{4} - 6276 p^{4} T^{6} + p^{8} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 58 T + 3839 T^{2} - 58 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 47 | $D_4\times C_2$ | \( 1 - 4272 T^{2} + 9382658 T^{4} - 4272 p^{4} T^{6} + p^{8} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 8960 T^{2} + 34984034 T^{4} - 8960 p^{4} T^{6} + p^{8} T^{8} \) |
| 59 | $D_4\times C_2$ | \( 1 - 6840 T^{2} + 30991922 T^{4} - 6840 p^{4} T^{6} + p^{8} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 40 T + 5574 T^{2} + 40 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 67 | $D_{4}$ | \( ( 1 - 108 T + 6791 T^{2} - 108 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 71 | $D_4\times C_2$ | \( 1 - 3276 T^{2} - 12941521 T^{4} - 3276 p^{4} T^{6} + p^{8} T^{8} \) |
| 73 | $D_{4}$ | \( ( 1 + 150 T + 16108 T^{2} + 150 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 79 | $D_{4}$ | \( ( 1 + 200 T + 22139 T^{2} + 200 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 18692 T^{2} + 182016950 T^{4} - 18692 p^{4} T^{6} + p^{8} T^{8} \) |
| 89 | $D_4\times C_2$ | \( 1 + 1008 T^{2} + 115251266 T^{4} + 1008 p^{4} T^{6} + p^{8} T^{8} \) |
| 97 | $D_{4}$ | \( ( 1 - 36 T + 13654 T^{2} - 36 p^{2} T^{3} + p^{4} T^{4} )^{2} \) |
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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
−5.79102211252777953358601089034, −5.68363869374666941326551778747, −5.67632317310966218186926154204, −5.33171293001931046184858915227, −5.12092919705318355311538786510, −5.08744298881667388309998786324, −4.72922748802880580921930214289, −4.62612792162764018032784401743, −4.27063365342100212400430576366, −4.22795510829856981420319457907, −3.82337761266811092016555880402, −3.67021604602178012522855661075, −3.33390868946035856644329841779, −3.22921605627760676297592645180, −3.05941640484130257429104734311, −2.93094268215750174166671715473, −2.57948675758706902544691584389, −2.37050636178684380213507036993, −1.84608061869725282769178080941, −1.49108780204002534705367932769, −1.46326666346584316021645984702, −1.11292861951753441717489165645, −0.912967803661233334172730827587, −0.49111395396322473496093690781, −0.27340423764891176012295353630,
0.27340423764891176012295353630, 0.49111395396322473496093690781, 0.912967803661233334172730827587, 1.11292861951753441717489165645, 1.46326666346584316021645984702, 1.49108780204002534705367932769, 1.84608061869725282769178080941, 2.37050636178684380213507036993, 2.57948675758706902544691584389, 2.93094268215750174166671715473, 3.05941640484130257429104734311, 3.22921605627760676297592645180, 3.33390868946035856644329841779, 3.67021604602178012522855661075, 3.82337761266811092016555880402, 4.22795510829856981420319457907, 4.27063365342100212400430576366, 4.62612792162764018032784401743, 4.72922748802880580921930214289, 5.08744298881667388309998786324, 5.12092919705318355311538786510, 5.33171293001931046184858915227, 5.67632317310966218186926154204, 5.68363869374666941326551778747, 5.79102211252777953358601089034
