| L(s) = 1 | − 2·2-s + 2·3-s + 4-s − 4·6-s − 4·7-s + 3·9-s + 2·12-s + 8·14-s + 16-s − 8·17-s − 6·18-s + 8·19-s − 8·21-s + 8·23-s + 4·27-s − 4·28-s + 4·29-s + 2·32-s + 16·34-s + 3·36-s − 12·37-s − 16·38-s − 4·41-s + 16·42-s − 12·43-s − 16·46-s + 8·47-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 1.15·3-s + 1/2·4-s − 1.63·6-s − 1.51·7-s + 9-s + 0.577·12-s + 2.13·14-s + 1/4·16-s − 1.94·17-s − 1.41·18-s + 1.83·19-s − 1.74·21-s + 1.66·23-s + 0.769·27-s − 0.755·28-s + 0.742·29-s + 0.353·32-s + 2.74·34-s + 1/2·36-s − 1.97·37-s − 2.59·38-s − 0.624·41-s + 2.46·42-s − 1.82·43-s − 2.35·46-s + 1.16·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 82355625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 3 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_4$ | \( 1 + 4 T + 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 8 T + 42 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 8 T + 46 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 + 12 T + 78 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 4 T + 78 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 12 T + 114 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 53 | $D_{4}$ | \( 1 - 4 T - 18 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 12 T + 126 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 102 T^{2} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 16 T + 174 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 86 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 4 T + 150 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 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
−7.45626179333232603013097331723, −7.23375731626928246649683900791, −7.10774106834559448868324109018, −6.64440319528900769632641712493, −6.53815252835026589714107668188, −6.07132353966100590054304636200, −5.36834097993701678483862804189, −5.18974692350181260261735951072, −4.82835635683543541720256987534, −4.29507947787404545375897274359, −3.83535976864819732693744540151, −3.43966367862723552776864509876, −3.21264429090138532063682230885, −2.76074017464105235179380387766, −2.51541572512482221651792551906, −1.95463227545759355280640341881, −1.17186500720464509088722758588, −1.09550939335707831457253706998, 0, 0,
1.09550939335707831457253706998, 1.17186500720464509088722758588, 1.95463227545759355280640341881, 2.51541572512482221651792551906, 2.76074017464105235179380387766, 3.21264429090138532063682230885, 3.43966367862723552776864509876, 3.83535976864819732693744540151, 4.29507947787404545375897274359, 4.82835635683543541720256987534, 5.18974692350181260261735951072, 5.36834097993701678483862804189, 6.07132353966100590054304636200, 6.53815252835026589714107668188, 6.64440319528900769632641712493, 7.10774106834559448868324109018, 7.23375731626928246649683900791, 7.45626179333232603013097331723
