| L(s) = 1 | − 4·2-s + 10·4-s − 20·8-s − 5·9-s − 10·11-s + 35·16-s + 20·18-s + 40·22-s − 15·25-s + 4·29-s − 56·32-s − 50·36-s + 12·37-s + 6·43-s − 100·44-s + 60·50-s − 38·53-s − 16·58-s + 84·64-s − 36·67-s − 4·71-s + 100·72-s − 48·74-s − 10·79-s + 5·81-s − 24·86-s + 200·88-s + ⋯ |
| L(s) = 1 | − 2.82·2-s + 5·4-s − 7.07·8-s − 5/3·9-s − 3.01·11-s + 35/4·16-s + 4.71·18-s + 8.52·22-s − 3·25-s + 0.742·29-s − 9.89·32-s − 8.33·36-s + 1.97·37-s + 0.914·43-s − 15.0·44-s + 8.48·50-s − 5.21·53-s − 2.10·58-s + 21/2·64-s − 4.39·67-s − 0.474·71-s + 11.7·72-s − 5.57·74-s − 1.12·79-s + 5/9·81-s − 2.58·86-s + 21.3·88-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 7^{8} \cdot 29^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{4} \) |
| 7 | | \( 1 \) |
| 29 | $C_1$ | \( ( 1 - T )^{4} \) |
| good | 3 | $C_2^2:C_4$ | \( 1 + 5 T^{2} + 20 T^{4} + 5 p^{2} T^{6} + p^{4} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 + 3 p T^{2} + 102 T^{4} + 3 p^{3} T^{6} + p^{4} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 5 T + 24 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 + 15 T^{2} + 30 p T^{4} + 15 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 + 22 T^{2} + 682 T^{4} + 22 p^{2} T^{6} + p^{4} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 16 T^{2} + 718 T^{4} - 16 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 22 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 + 87 T^{2} + 3810 T^{4} + 87 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 66 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 + 118 T^{2} + 6826 T^{4} + 118 p^{2} T^{6} + p^{4} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 183 T^{2} + 12786 T^{4} + 183 p^{2} T^{6} + p^{4} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 19 T + 192 T^{2} + 19 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 + 190 T^{2} + 15970 T^{4} + 190 p^{2} T^{6} + p^{4} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 + 224 T^{2} + 19918 T^{4} + 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 18 T + 198 T^{2} + 18 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 2 T - 10 T^{2} + 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 58 T^{2} + 874 T^{4} - 58 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 5 T + 58 T^{2} + 5 p T^{3} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 + 206 T^{2} + 23010 T^{4} + 206 p^{2} T^{6} + p^{4} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 + 266 T^{2} + 32154 T^{4} + 266 p^{2} T^{6} + p^{4} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 150 T^{2} + 19530 T^{4} + 150 p^{2} T^{6} + p^{4} T^{8} \) |
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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
−6.52693788744558533786673309648, −6.36887193651266158540024653883, −6.21505931494885762205184734012, −6.19764330858997277098079741178, −5.99480339374861965606887238176, −5.67913894031928254313598443786, −5.54820202592595056797728821640, −5.37020872563565395947631442802, −5.29943555319643442347434210663, −4.63603661540254214533212858966, −4.59272988266234706167651728090, −4.56934616039206357546226241080, −4.21285481231937893915527427589, −3.67279581643659516633387211283, −3.38312255693820529430211035809, −3.34918860537743190165069392337, −2.96355715608654607249654119842, −2.67473567460688958214678111473, −2.67333890855701906596824097255, −2.50321110819973847478567922670, −2.26700050458092135828978076196, −1.76244192578826469777742681651, −1.67810418515543514214336499932, −1.20001819873064498535440663765, −1.06105388899107994111568830543, 0, 0, 0, 0,
1.06105388899107994111568830543, 1.20001819873064498535440663765, 1.67810418515543514214336499932, 1.76244192578826469777742681651, 2.26700050458092135828978076196, 2.50321110819973847478567922670, 2.67333890855701906596824097255, 2.67473567460688958214678111473, 2.96355715608654607249654119842, 3.34918860537743190165069392337, 3.38312255693820529430211035809, 3.67279581643659516633387211283, 4.21285481231937893915527427589, 4.56934616039206357546226241080, 4.59272988266234706167651728090, 4.63603661540254214533212858966, 5.29943555319643442347434210663, 5.37020872563565395947631442802, 5.54820202592595056797728821640, 5.67913894031928254313598443786, 5.99480339374861965606887238176, 6.19764330858997277098079741178, 6.21505931494885762205184734012, 6.36887193651266158540024653883, 6.52693788744558533786673309648
