| L(s) = 1 | + 2·3-s − 2·5-s − 9-s − 4·11-s + 4·13-s − 4·15-s − 4·17-s + 2·23-s + 3·25-s − 6·27-s − 2·29-s − 12·31-s − 8·33-s + 8·39-s + 10·41-s − 10·43-s + 2·45-s + 4·47-s − 8·51-s − 8·53-s + 8·55-s − 8·59-s + 6·61-s − 8·65-s − 22·67-s + 4·69-s + 8·71-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 1/3·9-s − 1.20·11-s + 1.10·13-s − 1.03·15-s − 0.970·17-s + 0.417·23-s + 3/5·25-s − 1.15·27-s − 0.371·29-s − 2.15·31-s − 1.39·33-s + 1.28·39-s + 1.56·41-s − 1.52·43-s + 0.298·45-s + 0.583·47-s − 1.12·51-s − 1.09·53-s + 1.07·55-s − 1.04·59-s + 0.768·61-s − 0.992·65-s − 2.68·67-s + 0.481·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15366400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 2 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 30 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 45 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 41 | $D_{4}$ | \( 1 - 10 T + 99 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 10 T + 109 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 + 8 T + 114 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 8 T + 62 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 59 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 22 T + 253 T^{2} + 22 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 8 T + 86 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 142 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 24 T + 294 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 5 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 6 T + 155 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 12 T + 198 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
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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
−8.145898182733330954121055170249, −8.119928995173777812942781761215, −7.58566790301385592765185910177, −7.40639400107566347900685862524, −6.97629223454409264378965851240, −6.51338787032861103155137395659, −5.94223429253028487084607277618, −5.80010369114703331750718807753, −5.17009601974616207308866980103, −4.97011694112183452563035874189, −4.18356589453002750057446732122, −4.11262001685906042033959309059, −3.50507988350979725692667066838, −3.22173237827856444670520136259, −2.66649123858515447411599559523, −2.56485587577274279809071502502, −1.74495896908886396167095089638, −1.31466325839523224267556805540, 0, 0,
1.31466325839523224267556805540, 1.74495896908886396167095089638, 2.56485587577274279809071502502, 2.66649123858515447411599559523, 3.22173237827856444670520136259, 3.50507988350979725692667066838, 4.11262001685906042033959309059, 4.18356589453002750057446732122, 4.97011694112183452563035874189, 5.17009601974616207308866980103, 5.80010369114703331750718807753, 5.94223429253028487084607277618, 6.51338787032861103155137395659, 6.97629223454409264378965851240, 7.40639400107566347900685862524, 7.58566790301385592765185910177, 8.119928995173777812942781761215, 8.145898182733330954121055170249
