| L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 8·5-s − 4·6-s + 2·7-s + 2·8-s + 9-s + 16·10-s − 11-s + 2·12-s − 6·13-s − 4·14-s − 16·15-s − 4·16-s − 10·17-s − 2·18-s + 5·19-s − 8·20-s + 4·21-s + 2·22-s + 5·23-s + 4·24-s + 20·25-s + 12·26-s − 2·27-s + 2·28-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s − 3.57·5-s − 1.63·6-s + 0.755·7-s + 0.707·8-s + 1/3·9-s + 5.05·10-s − 0.301·11-s + 0.577·12-s − 1.66·13-s − 1.06·14-s − 4.13·15-s − 16-s − 2.42·17-s − 0.471·18-s + 1.14·19-s − 1.78·20-s + 0.872·21-s + 0.426·22-s + 1.04·23-s + 0.816·24-s + 4·25-s + 2.35·26-s − 0.384·27-s + 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{4} \cdot 7^{4} \cdot 13^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1257338870\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1257338870\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{4} \) |
| 11 | $C_2$$\times$$C_2^2$ | \( ( 1 + T + p T^{2} )^{2}( 1 - T - 10 T^{2} - p T^{3} + p^{2} T^{4} ) \) |
| 17 | $C_2^2$ | \( ( 1 + 5 T + 8 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $D_4\times C_2$ | \( 1 - 5 T + 13 T^{2} + 130 T^{3} - 692 T^{4} + 130 p T^{5} + 13 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $D_4\times C_2$ | \( 1 - 5 T + 5 T^{2} + 130 T^{3} - 704 T^{4} + 130 p T^{5} + 5 p^{2} T^{6} - 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $D_4\times C_2$ | \( 1 + 5 T - 7 T^{2} - 130 T^{3} - 542 T^{4} - 130 p T^{5} - 7 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $D_{4}$ | \( ( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 2 T - 33 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_4\times C_2$ | \( 1 - 3 T - 43 T^{2} + 90 T^{3} + 654 T^{4} + 90 p T^{5} - 43 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $D_4\times C_2$ | \( 1 + 13 T + 73 T^{2} + 130 T^{3} + 100 T^{4} + 130 p T^{5} + 73 p^{2} T^{6} + 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $D_{4}$ | \( ( 1 + 3 T + 64 T^{2} + 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{4} \) |
| 59 | $D_4\times C_2$ | \( 1 - 13 T + 41 T^{2} - 130 T^{3} + 2932 T^{4} - 130 p T^{5} + 41 p^{2} T^{6} - 13 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2^2$ | \( ( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 19 T + 169 T^{2} - 1102 T^{3} + 8188 T^{4} - 1102 p T^{5} + 169 p^{2} T^{6} - 19 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $D_4\times C_2$ | \( 1 + 7 T - 73 T^{2} - 140 T^{3} + 6364 T^{4} - 140 p T^{5} - 73 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{4} \) |
| 79 | $D_{4}$ | \( ( 1 - 15 T + 182 T^{2} - 15 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - T + 134 T^{2} - p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $C_2^2$ | \( ( 1 + 9 T - 8 T^{2} + 9 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 + 2 T - 62 T^{2} - 256 T^{3} - 5441 T^{4} - 256 p T^{5} - 62 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
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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
−8.011129274506011085100366665662, −7.76659180068789645909828552232, −7.49775028936306690129423511437, −7.23548071869529114342173401628, −7.09215554927866160353667894650, −6.90302359436369252990671801839, −6.64646963877933473075456209287, −6.49743424848565371082611113071, −5.63110288819011479644914281874, −5.56568902393654669087316917983, −5.04419061149606610582208603161, −4.94494751860031872216306115013, −4.85979907054153562112379120118, −4.33411656556602806139700567678, −4.11893531363589770482950566249, −3.97482456923709664462515045886, −3.59032881975647066795230539223, −3.41126679667870522850116032338, −3.26423483179434402926065797716, −2.51635319766143103932682499058, −2.39011307957279222375653869731, −1.90439736054073195820049045610, −1.66152350960272933531137111645, −0.58423465200281592779347530177, −0.23682064380232302450242243318,
0.23682064380232302450242243318, 0.58423465200281592779347530177, 1.66152350960272933531137111645, 1.90439736054073195820049045610, 2.39011307957279222375653869731, 2.51635319766143103932682499058, 3.26423483179434402926065797716, 3.41126679667870522850116032338, 3.59032881975647066795230539223, 3.97482456923709664462515045886, 4.11893531363589770482950566249, 4.33411656556602806139700567678, 4.85979907054153562112379120118, 4.94494751860031872216306115013, 5.04419061149606610582208603161, 5.56568902393654669087316917983, 5.63110288819011479644914281874, 6.49743424848565371082611113071, 6.64646963877933473075456209287, 6.90302359436369252990671801839, 7.09215554927866160353667894650, 7.23548071869529114342173401628, 7.49775028936306690129423511437, 7.76659180068789645909828552232, 8.011129274506011085100366665662
