| 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
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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
−6.40719679974778428091616346803, −6.06102041841345672151865202214, −5.76106331876439787359830040193, −5.69817207617283716843738616796, −5.59640980200145906513122209379, −5.09780472053480442004092691627, −4.97160487931646326432183391539, −4.77710627583743821047373331463, −4.44453780010148318864540208974, −4.08850333025252939707504988480, −4.03401780231672316550553203568, −3.92146802954181133914315874624, −3.85059115421446940163426821329, −3.27400550346832467431598865267, −3.24678253448700997887078564243, −3.01811589386946244167344807599, −2.78788152246654881486903211543, −2.35753256529048591859711680504, −2.24591202584379325525133125615, −2.21386431989342816423884232724, −2.20415261420380192181012202530, −1.66309903040462948569573927499, −1.47207023580028135728887638537, −1.31681444181773817908752478313, −1.21480611825906003316691154228, 0, 0, 0, 0,
1.21480611825906003316691154228, 1.31681444181773817908752478313, 1.47207023580028135728887638537, 1.66309903040462948569573927499, 2.20415261420380192181012202530, 2.21386431989342816423884232724, 2.24591202584379325525133125615, 2.35753256529048591859711680504, 2.78788152246654881486903211543, 3.01811589386946244167344807599, 3.24678253448700997887078564243, 3.27400550346832467431598865267, 3.85059115421446940163426821329, 3.92146802954181133914315874624, 4.03401780231672316550553203568, 4.08850333025252939707504988480, 4.44453780010148318864540208974, 4.77710627583743821047373331463, 4.97160487931646326432183391539, 5.09780472053480442004092691627, 5.59640980200145906513122209379, 5.69817207617283716843738616796, 5.76106331876439787359830040193, 6.06102041841345672151865202214, 6.40719679974778428091616346803
