| L(s) = 1 | + 2-s − 5-s + 7-s − 8-s − 10-s − 5·13-s + 14-s − 16-s − 12·17-s + 10·19-s − 3·23-s − 5·26-s − 8·31-s − 12·34-s − 35-s + 4·37-s + 10·38-s + 40-s − 3·41-s + 4·43-s − 3·46-s − 9·47-s + 7·49-s − 18·53-s − 56-s − 15·59-s + 4·61-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.447·5-s + 0.377·7-s − 0.353·8-s − 0.316·10-s − 1.38·13-s + 0.267·14-s − 1/4·16-s − 2.91·17-s + 2.29·19-s − 0.625·23-s − 0.980·26-s − 1.43·31-s − 2.05·34-s − 0.169·35-s + 0.657·37-s + 1.62·38-s + 0.158·40-s − 0.468·41-s + 0.609·43-s − 0.442·46-s − 1.31·47-s + 49-s − 2.47·53-s − 0.133·56-s − 1.95·59-s + 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 656100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.493391293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.493391293\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T + 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 3 T - 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 9 T + 34 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 4 T - 45 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 4 T - 51 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 14 T + 117 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 6 T - 47 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 18 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−10.71923736381369870936756785331, −9.894136239434421581607890412244, −9.513796483003927778005201441316, −9.257339951327747561909550095393, −8.954891634162904688562474182502, −8.149809521902787185100827241656, −7.890922550572815638136302182455, −7.42561644171098739360614113734, −7.05870347507207686627421478004, −6.44494177846937143488466013796, −6.20066908148836744066907098762, −5.25552357429392814403947732813, −5.15907863547186272717023210263, −4.47428517403903172147137115732, −4.42946551313378878683673142185, −3.35472269097484007578400158635, −3.32018167511013197192716409628, −2.14426198828478723991062551941, −2.05743060299448637290510215638, −0.52064783038718887033954295515,
0.52064783038718887033954295515, 2.05743060299448637290510215638, 2.14426198828478723991062551941, 3.32018167511013197192716409628, 3.35472269097484007578400158635, 4.42946551313378878683673142185, 4.47428517403903172147137115732, 5.15907863547186272717023210263, 5.25552357429392814403947732813, 6.20066908148836744066907098762, 6.44494177846937143488466013796, 7.05870347507207686627421478004, 7.42561644171098739360614113734, 7.890922550572815638136302182455, 8.149809521902787185100827241656, 8.954891634162904688562474182502, 9.257339951327747561909550095393, 9.513796483003927778005201441316, 9.894136239434421581607890412244, 10.71923736381369870936756785331
