| L(s) = 1 | + 4·2-s + 2·3-s + 10·4-s − 2·5-s + 8·6-s + 3·7-s + 20·8-s + 4·9-s − 8·10-s + 20·12-s + 13-s + 12·14-s − 4·15-s + 35·16-s − 7·17-s + 16·18-s + 4·19-s − 20·20-s + 6·21-s − 5·23-s + 40·24-s − 4·25-s + 4·26-s + 6·27-s + 30·28-s − 14·29-s − 16·30-s + ⋯ |
| L(s) = 1 | + 2.82·2-s + 1.15·3-s + 5·4-s − 0.894·5-s + 3.26·6-s + 1.13·7-s + 7.07·8-s + 4/3·9-s − 2.52·10-s + 5.77·12-s + 0.277·13-s + 3.20·14-s − 1.03·15-s + 35/4·16-s − 1.69·17-s + 3.77·18-s + 0.917·19-s − 4.47·20-s + 1.30·21-s − 1.04·23-s + 8.16·24-s − 4/5·25-s + 0.784·26-s + 1.15·27-s + 5.66·28-s − 2.59·29-s − 2.92·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 11^{8} \cdot 19^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(61.38798706\) |
| \(L(\frac12)\) |
\(\approx\) |
\(61.38798706\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{4} \) |
| 11 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2 \wr S_4$ | \( 1 - 2 T + 2 T^{3} - T^{4} + 2 p T^{5} - 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 5 | $C_2 \wr S_4$ | \( 1 + 2 T + 8 T^{2} + 24 T^{3} + 54 T^{4} + 24 p T^{5} + 8 p^{2} T^{6} + 2 p^{3} T^{7} + p^{4} T^{8} \) |
| 7 | $C_2 \wr S_4$ | \( 1 - 3 T + p T^{2} + 4 T^{3} - 48 T^{4} + 4 p T^{5} + p^{3} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 13 | $C_2 \wr S_4$ | \( 1 - T + 22 T^{2} - 59 T^{3} + 2 p^{2} T^{4} - 59 p T^{5} + 22 p^{2} T^{6} - p^{3} T^{7} + p^{4} T^{8} \) |
| 17 | $C_2 \wr S_4$ | \( 1 + 7 T + 62 T^{2} + 261 T^{3} + 1398 T^{4} + 261 p T^{5} + 62 p^{2} T^{6} + 7 p^{3} T^{7} + p^{4} T^{8} \) |
| 23 | $C_2 \wr S_4$ | \( 1 + 5 T + 71 T^{2} + 312 T^{3} + 2304 T^{4} + 312 p T^{5} + 71 p^{2} T^{6} + 5 p^{3} T^{7} + p^{4} T^{8} \) |
| 29 | $C_2 \wr S_4$ | \( 1 + 14 T + 146 T^{2} + 1134 T^{3} + 6717 T^{4} + 1134 p T^{5} + 146 p^{2} T^{6} + 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 31 | $C_2 \wr S_4$ | \( 1 - 4 T + 76 T^{2} - 212 T^{3} + 3062 T^{4} - 212 p T^{5} + 76 p^{2} T^{6} - 4 p^{3} T^{7} + p^{4} T^{8} \) |
| 37 | $C_2 \wr S_4$ | \( 1 - 12 T + 70 T^{2} + 118 T^{3} - 2127 T^{4} + 118 p T^{5} + 70 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 41 | $C_2 \wr S_4$ | \( 1 - 12 T + 146 T^{2} - 966 T^{3} + 7638 T^{4} - 966 p T^{5} + 146 p^{2} T^{6} - 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 43 | $C_2 \wr S_4$ | \( 1 - 14 T + 52 T^{2} + 34 T^{3} - 14 p T^{4} + 34 p T^{5} + 52 p^{2} T^{6} - 14 p^{3} T^{7} + p^{4} T^{8} \) |
| 47 | $C_2 \wr S_4$ | \( 1 + 12 T + 206 T^{2} + 1524 T^{3} + 14595 T^{4} + 1524 p T^{5} + 206 p^{2} T^{6} + 12 p^{3} T^{7} + p^{4} T^{8} \) |
| 53 | $C_2 \wr S_4$ | \( 1 + 10 T + 206 T^{2} + 1386 T^{3} + 16005 T^{4} + 1386 p T^{5} + 206 p^{2} T^{6} + 10 p^{3} T^{7} + p^{4} T^{8} \) |
| 59 | $C_2 \wr S_4$ | \( 1 - 3 T + 191 T^{2} - 378 T^{3} + 15684 T^{4} - 378 p T^{5} + 191 p^{2} T^{6} - 3 p^{3} T^{7} + p^{4} T^{8} \) |
| 61 | $C_2 \wr S_4$ | \( 1 - 6 T + 112 T^{2} - 872 T^{3} + 8634 T^{4} - 872 p T^{5} + 112 p^{2} T^{6} - 6 p^{3} T^{7} + p^{4} T^{8} \) |
| 67 | $C_2 \wr S_4$ | \( 1 - 15 T + 184 T^{2} - 887 T^{3} + 7806 T^{4} - 887 p T^{5} + 184 p^{2} T^{6} - 15 p^{3} T^{7} + p^{4} T^{8} \) |
| 71 | $C_2 \wr S_4$ | \( 1 + 16 T + 230 T^{2} + 1950 T^{3} + 19662 T^{4} + 1950 p T^{5} + 230 p^{2} T^{6} + 16 p^{3} T^{7} + p^{4} T^{8} \) |
| 73 | $C_2 \wr S_4$ | \( 1 + 9 T + 298 T^{2} + 1939 T^{3} + 32814 T^{4} + 1939 p T^{5} + 298 p^{2} T^{6} + 9 p^{3} T^{7} + p^{4} T^{8} \) |
| 79 | $C_2 \wr S_4$ | \( 1 - 20 T + 394 T^{2} - 4706 T^{3} + 49678 T^{4} - 4706 p T^{5} + 394 p^{2} T^{6} - 20 p^{3} T^{7} + p^{4} T^{8} \) |
| 83 | $C_2 \wr S_4$ | \( 1 + 22 T + 500 T^{2} + 5988 T^{3} + 69714 T^{4} + 5988 p T^{5} + 500 p^{2} T^{6} + 22 p^{3} T^{7} + p^{4} T^{8} \) |
| 89 | $C_2 \wr S_4$ | \( 1 + 8 T + 278 T^{2} + 1854 T^{3} + 35094 T^{4} + 1854 p T^{5} + 278 p^{2} T^{6} + 8 p^{3} T^{7} + p^{4} T^{8} \) |
| 97 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{4} \) |
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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
−5.89461595572480936415458167533, −5.60262149052477330050583802751, −5.50189353697330139567086943314, −5.08379249782215090096281305567, −5.04373487579619737031195554456, −4.77589068985574244236515326314, −4.64066644005231217420585661219, −4.30777053859391735256563633832, −4.16772953930730639831984674315, −4.10337334805057890914148738095, −4.08004915013758772737223039029, −3.83660789915673565219901422352, −3.58736682015927767129952277095, −3.29391591670469208497471683845, −3.05583947662536031124272865837, −2.94394319010821853078777909271, −2.63201622760380838020872472529, −2.36265967121569850127761566252, −2.23598363123536398039563528308, −2.05656759438047773569047208272, −1.73277410303923656236999009676, −1.46626578204086222463185624527, −1.36105493288492662810314439805, −0.78044104114830551867073652068, −0.40387372729302096568988663116,
0.40387372729302096568988663116, 0.78044104114830551867073652068, 1.36105493288492662810314439805, 1.46626578204086222463185624527, 1.73277410303923656236999009676, 2.05656759438047773569047208272, 2.23598363123536398039563528308, 2.36265967121569850127761566252, 2.63201622760380838020872472529, 2.94394319010821853078777909271, 3.05583947662536031124272865837, 3.29391591670469208497471683845, 3.58736682015927767129952277095, 3.83660789915673565219901422352, 4.08004915013758772737223039029, 4.10337334805057890914148738095, 4.16772953930730639831984674315, 4.30777053859391735256563633832, 4.64066644005231217420585661219, 4.77589068985574244236515326314, 5.04373487579619737031195554456, 5.08379249782215090096281305567, 5.50189353697330139567086943314, 5.60262149052477330050583802751, 5.89461595572480936415458167533
