| L(s) = 1 | − 2·2-s − 2·3-s + 3·4-s + 4·6-s + 4·7-s − 4·8-s + 3·9-s − 6·12-s − 4·13-s − 8·14-s + 5·16-s − 8·17-s − 6·18-s + 4·19-s − 8·21-s + 2·23-s + 8·24-s + 8·26-s − 4·27-s + 12·28-s − 4·29-s + 12·31-s − 6·32-s + 16·34-s + 9·36-s − 4·37-s − 8·38-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 − 1.73·12-s − 1.10·13-s − 2.13·14-s + 5/4·16-s − 1.94·17-s − 1.41·18-s + 0.917·19-s − 1.74·21-s + 0.417·23-s + 1.63·24-s + 1.56·26-s − 0.769·27-s + 2.26·28-s − 0.742·29-s + 2.15·31-s − 1.06·32-s + 2.74·34-s + 3/2·36-s − 0.657·37-s − 1.29·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11902500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 23 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 4 T + 22 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 4 T + 24 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 4 T + 30 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 4 T + 76 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 8 T + 90 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 104 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 4 T + 90 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 24 T + 248 T^{2} + 24 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 46 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 4 T + 124 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 4 T + 136 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T + 118 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 4 T + 130 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 164 T^{2} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 24 T + 314 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−8.277055308079875554204520144244, −8.139417252978483039158086461813, −7.56845724493431727665088476758, −7.55695548737606722185206339050, −6.82802835120300659643134007627, −6.71530493635204233734222748307, −6.24071564106165129026177821930, −6.09229068560877972537183871214, −5.09526678448101388994915688245, −5.03254920212594118912116415425, −4.74602879821232676206647194535, −4.55855340168259282939843617172, −3.46715147162141545881031913478, −3.29999592043231135090551328894, −2.32636192782969840948137039778, −2.16335458216307610355650323753, −1.38595951800893395693136799332, −1.29272130638678353410036601907, 0, 0,
1.29272130638678353410036601907, 1.38595951800893395693136799332, 2.16335458216307610355650323753, 2.32636192782969840948137039778, 3.29999592043231135090551328894, 3.46715147162141545881031913478, 4.55855340168259282939843617172, 4.74602879821232676206647194535, 5.03254920212594118912116415425, 5.09526678448101388994915688245, 6.09229068560877972537183871214, 6.24071564106165129026177821930, 6.71530493635204233734222748307, 6.82802835120300659643134007627, 7.55695548737606722185206339050, 7.56845724493431727665088476758, 8.139417252978483039158086461813, 8.277055308079875554204520144244
