| L(s) = 1 | − 4·2-s − 4·3-s + 10·4-s + 16·6-s − 20·8-s + 10·9-s − 2·11-s − 40·12-s + 35·16-s + 10·17-s − 40·18-s + 4·19-s + 8·22-s − 5·23-s + 80·24-s − 8·25-s − 20·27-s − 3·29-s + 9·31-s − 56·32-s + 8·33-s − 40·34-s + 100·36-s − 14·37-s − 16·38-s + 4·41-s + 21·43-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 2.30·3-s + 5·4-s + 6.53·6-s − 7.07·8-s + 10/3·9-s − 0.603·11-s − 11.5·12-s + 35/4·16-s + 2.42·17-s − 9.42·18-s + 0.917·19-s + 1.70·22-s − 1.04·23-s + 16.3·24-s − 8/5·25-s − 3.84·27-s − 0.557·29-s + 1.61·31-s − 9.89·32-s + 1.39·33-s − 6.85·34-s + 50/3·36-s − 2.30·37-s − 2.59·38-s + 0.624·41-s + 3.20·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8458527037\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8458527037\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 3 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 5 | $S_4\times C_2$ | \( 1 + 8 T^{2} + 6 T^{3} + 51 T^{4} + 6 p T^{5} + 8 p^{2} T^{6} + p^{4} T^{8} \) |
| 11 | $C_2 \wr S_4$ | \( 1 + 2 T + 32 T^{2} + 4 p T^{3} + 469 T^{4} + 4 p^{2} T^{5} + 32 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 + 4 T^{2} - 48 T^{3} + 102 T^{4} - 48 p T^{5} + 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 - 10 T + 62 T^{2} - 256 T^{3} + 1078 T^{4} - 256 p T^{5} + 62 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 5 T + 32 T^{2} + 155 T^{3} + 1336 T^{4} + 155 p T^{5} + 32 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 3 T + 35 T^{2} + 126 T^{3} + 1644 T^{4} + 126 p T^{5} + 35 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 9 T + 91 T^{2} - 360 T^{3} + 2652 T^{4} - 360 p T^{5} + 91 p^{2} T^{6} - 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 + 14 T + 190 T^{2} + 1532 T^{3} + 11314 T^{4} + 1532 p T^{5} + 190 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 4 T + 68 T^{2} + 14 T^{3} + 1870 T^{4} + 14 p T^{5} + 68 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 21 T + 286 T^{2} - 2685 T^{3} + 20244 T^{4} - 2685 p T^{5} + 286 p^{2} T^{6} - 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 - 7 T + 164 T^{2} - 859 T^{3} + 11254 T^{4} - 859 p T^{5} + 164 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 7 T + 119 T^{2} + 694 T^{3} + 7972 T^{4} + 694 p T^{5} + 119 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 7 T + 53 T^{2} - 586 T^{3} + 8494 T^{4} - 586 p T^{5} + 53 p^{2} T^{6} - 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 23 T + 430 T^{2} - 4829 T^{3} + 45730 T^{4} - 4829 p T^{5} + 430 p^{2} T^{6} - 23 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 6 T + 28 T^{2} - 198 T^{3} + 2742 T^{4} - 198 p T^{5} + 28 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 - 2 T + 158 T^{2} - 380 T^{3} + 12382 T^{4} - 380 p T^{5} + 158 p^{2} T^{6} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 5 T + 106 T^{2} - 145 T^{3} + 3142 T^{4} - 145 p T^{5} + 106 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 + 11 T + 319 T^{2} + 2456 T^{3} + 38092 T^{4} + 2456 p T^{5} + 319 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 - 14 T + 272 T^{2} - 3200 T^{3} + 31417 T^{4} - 3200 p T^{5} + 272 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 - 10 T + 200 T^{2} - 8 p T^{3} + 14914 T^{4} - 8 p^{2} T^{5} + 200 p^{2} T^{6} - 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2 \wr S_4$ | \( 1 + T + 319 T^{2} + 232 T^{3} + 44116 T^{4} + 232 p T^{5} + 319 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
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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
−5.85059592794873442722624307545, −5.52016403888015839523414655578, −5.47462493426065146735221176690, −5.41460955225856559116393703934, −5.35355714971362669607760477029, −4.87453076045653853893701419983, −4.62776602917441108930849030494, −4.46287441055931450496799075880, −4.25828589944899078912211299922, −3.73914344264506685314323467025, −3.72127727765906776934097025754, −3.60968352552680732619742441402, −3.48866677007819485940150159308, −2.97045715824663710751863616726, −2.63722560392094282666601242514, −2.53943345156321214841740644385, −2.46452221540714075445432646705, −1.87325107224201813687647184864, −1.70703626830159924131556051716, −1.67534291264329115031398224968, −1.35122825452433132514110915833, −0.77426360056893783512585554240, −0.68652414094213973213125007465, −0.60887240030285224411112307281, −0.45559284860522707379766075676,
0.45559284860522707379766075676, 0.60887240030285224411112307281, 0.68652414094213973213125007465, 0.77426360056893783512585554240, 1.35122825452433132514110915833, 1.67534291264329115031398224968, 1.70703626830159924131556051716, 1.87325107224201813687647184864, 2.46452221540714075445432646705, 2.53943345156321214841740644385, 2.63722560392094282666601242514, 2.97045715824663710751863616726, 3.48866677007819485940150159308, 3.60968352552680732619742441402, 3.72127727765906776934097025754, 3.73914344264506685314323467025, 4.25828589944899078912211299922, 4.46287441055931450496799075880, 4.62776602917441108930849030494, 4.87453076045653853893701419983, 5.35355714971362669607760477029, 5.41460955225856559116393703934, 5.47462493426065146735221176690, 5.52016403888015839523414655578, 5.85059592794873442722624307545
