| L(s) = 1 | − 2·2-s − 5·4-s + 14·7-s + 12·8-s + 16·11-s + 76·13-s − 28·14-s − 11·16-s − 124·17-s − 96·19-s − 32·22-s − 16·23-s − 152·26-s − 70·28-s − 188·29-s − 120·31-s + 122·32-s + 248·34-s + 132·37-s + 192·38-s − 100·41-s + 536·43-s − 80·44-s + 32·46-s − 928·47-s + 147·49-s − 380·52-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 5/8·4-s + 0.755·7-s + 0.530·8-s + 0.438·11-s + 1.62·13-s − 0.534·14-s − 0.171·16-s − 1.76·17-s − 1.15·19-s − 0.310·22-s − 0.145·23-s − 1.14·26-s − 0.472·28-s − 1.20·29-s − 0.695·31-s + 0.673·32-s + 1.25·34-s + 0.586·37-s + 0.819·38-s − 0.380·41-s + 1.90·43-s − 0.274·44-s + 0.102·46-s − 2.88·47-s + 3/7·49-s − 1.01·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.599592602\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.599592602\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 2 | $D_{4}$ | \( 1 + p T + 9 T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 16 T - 474 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 76 T + 5806 T^{2} - 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 124 T + 13638 T^{2} + 124 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 96 T + 13974 T^{2} + 96 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 16 T + 15150 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 188 T + 34286 T^{2} + 188 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 120 T + 60590 T^{2} + 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 132 T + 70814 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 100 T + 89142 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 536 T + 200086 T^{2} - 536 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 928 T + 408830 T^{2} + 928 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 884 T + 460350 T^{2} - 884 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 104 T + 80534 T^{2} + 104 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 468 T + 494606 T^{2} + 468 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1688 T + 1302310 T^{2} - 1688 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 136 T + 540446 T^{2} - 136 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 508 T + 13078 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 432 T + 602142 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 584 T + 1172390 T^{2} + 584 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 1404 T + 1802390 T^{2} - 1404 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1188 T + 2161254 T^{2} - 1188 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.148862085010711906584867969920, −8.794681458740001325033186038992, −8.484545641305012890080494909306, −8.413142498911376468579965383861, −7.81737804189941971700052575388, −7.34273971571518094750399983007, −6.87848038655235803818136254248, −6.43627307088392360868036088376, −6.03663086875579142519159751243, −5.76120235415806023984537743350, −4.89630613255668499624331639274, −4.77491454911165300070185180077, −4.04163809891441802831668625980, −3.99789702734946826772808536767, −3.41717496467596497901784353303, −2.53948128548143492600136155468, −1.91194494073805489290818219797, −1.73025895600324539075960712269, −0.75739553855367824648032229375, −0.44771854296029226229800340617,
0.44771854296029226229800340617, 0.75739553855367824648032229375, 1.73025895600324539075960712269, 1.91194494073805489290818219797, 2.53948128548143492600136155468, 3.41717496467596497901784353303, 3.99789702734946826772808536767, 4.04163809891441802831668625980, 4.77491454911165300070185180077, 4.89630613255668499624331639274, 5.76120235415806023984537743350, 6.03663086875579142519159751243, 6.43627307088392360868036088376, 6.87848038655235803818136254248, 7.34273971571518094750399983007, 7.81737804189941971700052575388, 8.413142498911376468579965383861, 8.484545641305012890080494909306, 8.794681458740001325033186038992, 9.148862085010711906584867969920
