L(s) = 1 | + 2·2-s + 4-s − 2·5-s − 4·7-s − 4·9-s − 4·10-s − 4·11-s − 8·14-s + 16-s − 4·17-s − 8·18-s − 4·19-s − 2·20-s − 8·22-s + 3·25-s − 4·28-s − 12·31-s − 2·32-s − 8·34-s + 8·35-s − 4·36-s − 8·38-s + 12·41-s − 8·43-s − 4·44-s + 8·45-s + 4·47-s + ⋯ |
L(s) = 1 | + 1.41·2-s + 1/2·4-s − 0.894·5-s − 1.51·7-s − 4/3·9-s − 1.26·10-s − 1.20·11-s − 2.13·14-s + 1/4·16-s − 0.970·17-s − 1.88·18-s − 0.917·19-s − 0.447·20-s − 1.70·22-s + 3/5·25-s − 0.755·28-s − 2.15·31-s − 0.353·32-s − 1.37·34-s + 1.35·35-s − 2/3·36-s − 1.29·38-s + 1.87·41-s − 1.21·43-s − 0.603·44-s + 1.19·45-s + 0.583·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 714025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | | \( 1 \) |
good | 2 | $D_{4}$ | \( 1 - p T + 3 T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 24 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 4 T + 40 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 12 T + 80 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 12 T + 110 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 8 T + 52 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 4 T + 90 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 70 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 12 T + 136 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 + 4 T + 48 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 74 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 12 T + 194 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 166 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−10.15767672223446692403396422962, −9.367919148790405954684299632370, −9.026942848136500443879317012396, −8.964072752604296114631071938061, −8.173790422426284434131822630557, −7.64515130132572478231498067471, −7.58237203867389184223307986446, −6.73744656346224362208410858969, −6.36515192198646858037623592060, −5.97031552585204471714607698511, −5.46396933825867778629999014387, −5.09776968870667565798324370254, −4.38993345435487524521668471709, −4.25183294641275271894156570825, −3.33790417930147427701901169465, −3.30964100219849301181980737041, −2.69989485395194313129565717715, −1.96652415285945947577038679902, 0, 0,
1.96652415285945947577038679902, 2.69989485395194313129565717715, 3.30964100219849301181980737041, 3.33790417930147427701901169465, 4.25183294641275271894156570825, 4.38993345435487524521668471709, 5.09776968870667565798324370254, 5.46396933825867778629999014387, 5.97031552585204471714607698511, 6.36515192198646858037623592060, 6.73744656346224362208410858969, 7.58237203867389184223307986446, 7.64515130132572478231498067471, 8.173790422426284434131822630557, 8.964072752604296114631071938061, 9.026942848136500443879317012396, 9.367919148790405954684299632370, 10.15767672223446692403396422962