| L(s) = 1 | − 2-s − 3-s − 6·5-s + 6-s + 2·7-s + 8-s + 6·10-s + 6·11-s − 2·14-s + 6·15-s − 16-s + 3·17-s + 2·19-s − 2·21-s − 6·22-s + 6·23-s − 24-s + 17·25-s + 27-s − 3·29-s − 6·30-s + 8·31-s − 6·33-s − 3·34-s − 12·35-s − 7·37-s − 2·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s − 2.68·5-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1.89·10-s + 1.80·11-s − 0.534·14-s + 1.54·15-s − 1/4·16-s + 0.727·17-s + 0.458·19-s − 0.436·21-s − 1.27·22-s + 1.25·23-s − 0.204·24-s + 17/5·25-s + 0.192·27-s − 0.557·29-s − 1.09·30-s + 1.43·31-s − 1.04·33-s − 0.514·34-s − 2.02·35-s − 1.15·37-s − 0.324·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1028196 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8204895162\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8204895162\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 3 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 6 T + 13 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 3 T - 20 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 10 T + 57 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 6 T - 35 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 18 T + 235 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−10.17098017482769254764825271993, −9.749759990523999846255781611579, −9.020612880436558334322838123225, −9.001977840954471147593601840267, −8.466335345396784134129871665618, −7.943477552843235372051692402466, −7.83575555622262577170252828989, −7.40517446390885702761783735686, −6.81881506682350185880462802372, −6.71231790152800578131661867995, −5.98771638121019862726517837465, −5.11876807590698223064728580114, −5.01211841824957368727230080658, −4.27192319431339598584312167493, −4.03299847662459628965932896770, −3.48852194824463752075217887845, −3.16747251500587658017099786223, −1.94859706041883630553691688647, −0.965010919160016085248344419349, −0.68828884815438676489826169522,
0.68828884815438676489826169522, 0.965010919160016085248344419349, 1.94859706041883630553691688647, 3.16747251500587658017099786223, 3.48852194824463752075217887845, 4.03299847662459628965932896770, 4.27192319431339598584312167493, 5.01211841824957368727230080658, 5.11876807590698223064728580114, 5.98771638121019862726517837465, 6.71231790152800578131661867995, 6.81881506682350185880462802372, 7.40517446390885702761783735686, 7.83575555622262577170252828989, 7.943477552843235372051692402466, 8.466335345396784134129871665618, 9.001977840954471147593601840267, 9.020612880436558334322838123225, 9.749759990523999846255781611579, 10.17098017482769254764825271993
