| L(s) = 1 | + 2-s − 7·4-s − 16·5-s − 2·7-s − 7·8-s − 16·10-s + 76·13-s − 2·14-s − 7·16-s − 26·17-s + 54·19-s + 112·20-s − 224·23-s + 74·25-s + 76·26-s + 14·28-s + 222·29-s − 40·31-s − 71·32-s − 26·34-s + 32·35-s − 48·37-s + 54·38-s + 112·40-s − 494·41-s + 66·43-s − 224·46-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 7/8·4-s − 1.43·5-s − 0.107·7-s − 0.309·8-s − 0.505·10-s + 1.62·13-s − 0.0381·14-s − 0.109·16-s − 0.370·17-s + 0.652·19-s + 1.25·20-s − 2.03·23-s + 0.591·25-s + 0.573·26-s + 0.0944·28-s + 1.42·29-s − 0.231·31-s − 0.392·32-s − 0.131·34-s + 0.154·35-s − 0.213·37-s + 0.230·38-s + 0.442·40-s − 1.88·41-s + 0.234·43-s − 0.717·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1185921 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6255630125\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6255630125\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 11 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 - T + p^{3} T^{2} - p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + 16 T + 182 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 2 T + 654 T^{2} + 2 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 76 T + 5310 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 26 T + 2570 T^{2} + 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 54 T + 11774 T^{2} - 54 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 112 T + p^{3} T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 222 T + 43642 T^{2} - 222 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 40 T - 29250 T^{2} + 40 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 48 T + 85910 T^{2} + 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 494 T + 198818 T^{2} + 494 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 66 T + 99086 T^{2} - 66 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 64 T + 189662 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 84 T + 164350 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 196 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 1104 T + 736358 T^{2} - 1104 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 928 T + 626214 T^{2} - 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 456 T + 488494 T^{2} + 456 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 592 T + 341742 T^{2} - 592 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 230 T + 954126 T^{2} - 230 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 348 T + 307798 T^{2} - 348 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 972 T + 1645606 T^{2} + 972 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1184 T + 720510 T^{2} + 1184 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.714949415350422453951267600035, −9.223288690594318591865633609394, −8.570846636536188496700554724817, −8.549876990820262115259712868848, −8.106614811365882936076518514491, −7.83168455226693597105966904616, −7.23222088651524541053119347011, −6.62358137077237600788076064561, −6.43169156874888863544683200608, −5.87912704142991656078180717941, −5.18261944156241619139142558323, −4.96944369646958885102729316627, −4.27175487317162301759801363816, −4.01314321099030429025865365988, −3.52138135889255719812625422319, −3.38843235917156136407239053579, −2.40654320629661050181903592024, −1.69151501766364388446764586936, −0.923581783064526443548564789223, −0.23175505111287207800606330782,
0.23175505111287207800606330782, 0.923581783064526443548564789223, 1.69151501766364388446764586936, 2.40654320629661050181903592024, 3.38843235917156136407239053579, 3.52138135889255719812625422319, 4.01314321099030429025865365988, 4.27175487317162301759801363816, 4.96944369646958885102729316627, 5.18261944156241619139142558323, 5.87912704142991656078180717941, 6.43169156874888863544683200608, 6.62358137077237600788076064561, 7.23222088651524541053119347011, 7.83168455226693597105966904616, 8.106614811365882936076518514491, 8.549876990820262115259712868848, 8.570846636536188496700554724817, 9.223288690594318591865633609394, 9.714949415350422453951267600035
