| L(s) = 1 | + 2-s − 3-s − 3·4-s − 6-s − 7-s − 5·8-s − 11·11-s + 3·12-s − 7·13-s − 14-s − 12·17-s − 10·19-s + 21-s − 11·22-s + 8·23-s + 5·24-s − 7·26-s + 3·28-s + 6·29-s − 12·31-s + 9·32-s + 11·33-s − 12·34-s − 9·37-s − 10·38-s + 7·39-s − 3·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s − 3/2·4-s − 0.408·6-s − 0.377·7-s − 1.76·8-s − 3.31·11-s + 0.866·12-s − 1.94·13-s − 0.267·14-s − 2.91·17-s − 2.29·19-s + 0.218·21-s − 2.34·22-s + 1.66·23-s + 1.02·24-s − 1.37·26-s + 0.566·28-s + 1.11·29-s − 2.15·31-s + 1.59·32-s + 1.91·33-s − 2.05·34-s − 1.47·37-s − 1.62·38-s + 1.12·39-s − 0.468·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(3^{4} \cdot 5^{8} \cdot 11^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | 3 | $C_4$ | \( 1 + T + T^{2} + T^{3} + T^{4} \) |
| 5 | | \( 1 \) |
| 11 | $C_4$ | \( 1 + p T + 51 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| good | 2 | $C_4\times C_2$ | \( 1 - T + p^{2} T^{2} - p T^{3} + 9 T^{4} - p^{2} T^{5} + p^{4} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_4\times C_2$ | \( 1 + T - 6 T^{2} - 13 T^{3} + 29 T^{4} - 13 p T^{5} - 6 p^{2} T^{6} + p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2^2:C_4$ | \( 1 + 7 T + 6 T^{2} - 49 T^{3} - 181 T^{4} - 49 p T^{5} + 6 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 12 T + 37 T^{2} - 180 T^{3} - 1619 T^{4} - 180 p T^{5} + 37 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 + 10 T + 21 T^{2} - 70 T^{3} - 469 T^{4} - 70 p T^{5} + 21 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 4 T + 45 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_4\times C_2$ | \( 1 - 6 T + 7 T^{2} + 132 T^{3} - 995 T^{4} + 132 p T^{5} + 7 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2^2:C_4$ | \( 1 + 12 T + 63 T^{2} + 14 p T^{3} + 105 p T^{4} + 14 p^{2} T^{5} + 63 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2^2:C_4$ | \( 1 + 9 T - 6 T^{2} - 307 T^{3} - 1581 T^{4} - 307 p T^{5} - 6 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2^2:C_4$ | \( 1 + 3 T - 22 T^{2} + 171 T^{3} + 2215 T^{4} + 171 p T^{5} - 22 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2^2$ | \( ( 1 + 41 T^{2} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 17 T + 67 T^{2} - 335 T^{3} - 4344 T^{4} - 335 p T^{5} + 67 p^{2} T^{6} + 17 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2^2:C_4$ | \( 1 + 4 T - 47 T^{2} - 160 T^{3} + 2121 T^{4} - 160 p T^{5} - 47 p^{2} T^{6} + 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2^2:C_4$ | \( 1 + 6 T + 17 T^{2} - 222 T^{3} - 1685 T^{4} - 222 p T^{5} + 17 p^{2} T^{6} + 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 + 21 T + 245 T^{2} + 2559 T^{3} + 23404 T^{4} + 2559 p T^{5} + 245 p^{2} T^{6} + 21 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 - 3 T + 125 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2^2:C_4$ | \( 1 - 15 T + 119 T^{2} - 1455 T^{3} + 17296 T^{4} - 1455 p T^{5} + 119 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2^2:C_4$ | \( 1 + 14 T + 63 T^{2} + 850 T^{3} + 12521 T^{4} + 850 p T^{5} + 63 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_4\times C_2$ | \( 1 + 11 T + 42 T^{2} - 407 T^{3} - 7795 T^{4} - 407 p T^{5} + 42 p^{2} T^{6} + 11 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2^2:C_4$ | \( 1 + 13 T - 14 T^{2} - 421 T^{3} + 1449 T^{4} - 421 p T^{5} - 14 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 12 T + 209 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^2:C_4$ | \( 1 + 3 T - 43 T^{2} + 765 T^{3} + 11236 T^{4} + 765 p T^{5} - 43 p^{2} T^{6} + 3 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.065834432015334073595435861969, −7.34387920484458403009441156146, −7.29027267779659393532527895189, −7.10564800080218988309932375160, −6.86629372072709691780810932151, −6.56315553472296464045941603007, −6.53529626151337167468515020117, −6.12776646849369178144905722181, −5.85987782458731005950713473872, −5.49474566344415066909173765969, −5.25937232307213649253076028552, −5.18668297789487072108136564249, −5.02362418426370079843457011831, −4.74770029685664654720755678031, −4.54399372856039496822122012486, −4.36443232219653932078089323700, −4.23170214436569561634896961683, −3.80306157794372654404007056072, −3.43492264721405112112487596875, −2.92257934806023685130691371979, −2.73130761677589020497015689306, −2.60032446047226994380488686427, −2.48200418978685922423958999607, −1.79026773657428942826781064898, −1.64985787875352739664082176635, 0, 0, 0, 0,
1.64985787875352739664082176635, 1.79026773657428942826781064898, 2.48200418978685922423958999607, 2.60032446047226994380488686427, 2.73130761677589020497015689306, 2.92257934806023685130691371979, 3.43492264721405112112487596875, 3.80306157794372654404007056072, 4.23170214436569561634896961683, 4.36443232219653932078089323700, 4.54399372856039496822122012486, 4.74770029685664654720755678031, 5.02362418426370079843457011831, 5.18668297789487072108136564249, 5.25937232307213649253076028552, 5.49474566344415066909173765969, 5.85987782458731005950713473872, 6.12776646849369178144905722181, 6.53529626151337167468515020117, 6.56315553472296464045941603007, 6.86629372072709691780810932151, 7.10564800080218988309932375160, 7.29027267779659393532527895189, 7.34387920484458403009441156146, 8.065834432015334073595435861969
