| 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 + 3·13-s − 8·14-s + 5·16-s − 9·17-s + 6·18-s − 10·19-s − 8·21-s − 12·22-s − 12·23-s + 8·24-s + 6·26-s + 4·27-s − 12·28-s − 5·29-s − 6·31-s + 6·32-s − 12·33-s − 18·34-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 + 0.832·13-s − 2.13·14-s + 5/4·16-s − 2.18·17-s + 1.41·18-s − 2.29·19-s − 1.74·21-s − 2.55·22-s − 2.50·23-s + 1.63·24-s + 1.17·26-s + 0.769·27-s − 2.26·28-s − 0.928·29-s − 1.07·31-s + 1.06·32-s − 2.08·33-s − 3.08·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14062500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14062500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 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 | $D_{4}$ | \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 9 T + 43 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 10 T + 58 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 11 T + 93 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + T + 51 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 2 T + 82 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 14 T + 138 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 7 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 38 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 141 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 6 T + 98 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_4$ | \( 1 + 6 T + 26 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 13 T + 177 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 5 T + 183 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 21 T + 293 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
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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
−8.259633647654173618686785698789, −7.988174637716080447132848955378, −7.42815678841135281652092369507, −7.36611392450256727299891951985, −6.49394424038670159940513741366, −6.39710669728980340164309265892, −6.12689135219086971231296965488, −6.03617309650321930664057715823, −5.01402504278580237627984617784, −4.97397316065943581066659257845, −4.23696750095191986829772141844, −4.13628773024853578944184585156, −3.57448597375142721983323495056, −3.44511602104617996566977121073, −2.62873430187901273371634989070, −2.53144543610730095776689151034, −1.91476552505234852562012445836, −1.87174870722852918343160449398, 0, 0,
1.87174870722852918343160449398, 1.91476552505234852562012445836, 2.53144543610730095776689151034, 2.62873430187901273371634989070, 3.44511602104617996566977121073, 3.57448597375142721983323495056, 4.13628773024853578944184585156, 4.23696750095191986829772141844, 4.97397316065943581066659257845, 5.01402504278580237627984617784, 6.03617309650321930664057715823, 6.12689135219086971231296965488, 6.39710669728980340164309265892, 6.49394424038670159940513741366, 7.36611392450256727299891951985, 7.42815678841135281652092369507, 7.988174637716080447132848955378, 8.259633647654173618686785698789
