| L(s) = 1 | − 2·2-s + 2·3-s + 3·4-s − 4·6-s − 4·7-s − 4·8-s + 3·9-s + 6·11-s + 6·12-s + 8·14-s + 5·16-s − 6·18-s − 10·19-s − 8·21-s − 12·22-s − 8·23-s − 8·24-s + 4·27-s − 12·28-s + 8·29-s + 4·31-s − 6·32-s + 12·33-s + 9·36-s − 2·37-s + 20·38-s + 16·42-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 3/2·4-s − 1.63·6-s − 1.51·7-s − 1.41·8-s + 9-s + 1.80·11-s + 1.73·12-s + 2.13·14-s + 5/4·16-s − 1.41·18-s − 2.29·19-s − 1.74·21-s − 2.55·22-s − 1.66·23-s − 1.63·24-s + 0.769·27-s − 2.26·28-s + 1.48·29-s + 0.718·31-s − 1.06·32-s + 2.08·33-s + 3/2·36-s − 0.328·37-s + 3.24·38-s + 2.46·42-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 37 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 11 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 31 | $C_4$ | \( 1 - 4 T + 46 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 62 T^{2} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 102 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 58 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 16 T + 150 T^{2} + 16 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 4 T + 42 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 126 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 12 T + 150 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 - 4 T + 126 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 12 T + 162 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 142 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 4 T - 10 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 24 T + 302 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 78 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−8.220318208450092811089361484649, −7.956046526746712086804502987660, −7.19956747698195808478632959836, −6.83804010653734216558603780355, −6.55718424092204813713115804809, −6.54871684305188364908218830135, −6.13610009546469377052234514282, −5.85675795428365116250430764669, −4.81484771241578539528050335042, −4.67424431525943300128199132253, −3.94689417278836639800048134863, −3.83074465066149165186705324288, −3.34065535894603796168201984876, −2.99191726247056335718386863655, −2.43956209543189742603695032721, −2.11666826850686366909933546730, −1.44990813393888270720072076225, −1.29998578436147257363405025122, 0, 0,
1.29998578436147257363405025122, 1.44990813393888270720072076225, 2.11666826850686366909933546730, 2.43956209543189742603695032721, 2.99191726247056335718386863655, 3.34065535894603796168201984876, 3.83074465066149165186705324288, 3.94689417278836639800048134863, 4.67424431525943300128199132253, 4.81484771241578539528050335042, 5.85675795428365116250430764669, 6.13610009546469377052234514282, 6.54871684305188364908218830135, 6.55718424092204813713115804809, 6.83804010653734216558603780355, 7.19956747698195808478632959836, 7.956046526746712086804502987660, 8.220318208450092811089361484649
