| L(s) = 1 | + 2·2-s + 3·4-s − 2·5-s + 4·8-s − 4·10-s + 4·11-s − 4·13-s + 5·16-s − 4·17-s + 2·19-s − 6·20-s + 8·22-s + 6·23-s + 3·25-s − 8·26-s + 4·31-s + 6·32-s − 8·34-s + 4·37-s + 4·38-s − 8·40-s + 8·41-s + 12·43-s + 12·44-s + 12·46-s + 2·47-s + 6·50-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.894·5-s + 1.41·8-s − 1.26·10-s + 1.20·11-s − 1.10·13-s + 5/4·16-s − 0.970·17-s + 0.458·19-s − 1.34·20-s + 1.70·22-s + 1.25·23-s + 3/5·25-s − 1.56·26-s + 0.718·31-s + 1.06·32-s − 1.37·34-s + 0.657·37-s + 0.648·38-s − 1.26·40-s + 1.24·41-s + 1.82·43-s + 1.80·44-s + 1.76·46-s + 0.291·47-s + 0.848·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19448100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.113015523\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.113015523\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $D_{4}$ | \( 1 - 4 T + 19 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 23 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 11 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 4 T + 15 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 8 T + 91 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 12 T + 94 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 2 T - 17 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 10 T + 103 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 106 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 130 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 20 T + 206 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 16 T + 178 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 12 T + 166 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 20 T + 238 T^{2} - 20 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 8 T + 82 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.289223926293982750418183051504, −8.224612407374561864459874219950, −7.53324413696591049768438615888, −7.29225837033198205962304340408, −7.12825745597651647194803175703, −6.70391013670453982562341214437, −6.27399439007452498316047773998, −6.07146594367169852092749559929, −5.33487668508150270283435133838, −5.27765005433949840976188675200, −4.57389320587775153250067266461, −4.50829591839400497481880080537, −3.99269470374338941499724672115, −3.82526990085548992075904857288, −3.19152364950695690726965286511, −2.81362293862829108362201136848, −2.34586965416626740885020585846, −2.01952422189321673100542483409, −0.949663622582214501160397127911, −0.78465737339981128052290300713,
0.78465737339981128052290300713, 0.949663622582214501160397127911, 2.01952422189321673100542483409, 2.34586965416626740885020585846, 2.81362293862829108362201136848, 3.19152364950695690726965286511, 3.82526990085548992075904857288, 3.99269470374338941499724672115, 4.50829591839400497481880080537, 4.57389320587775153250067266461, 5.27765005433949840976188675200, 5.33487668508150270283435133838, 6.07146594367169852092749559929, 6.27399439007452498316047773998, 6.70391013670453982562341214437, 7.12825745597651647194803175703, 7.29225837033198205962304340408, 7.53324413696591049768438615888, 8.224612407374561864459874219950, 8.289223926293982750418183051504
