| L(s) = 1 | − 4·4-s − 6·11-s + 16·16-s − 64·19-s + 102·29-s + 166·31-s + 216·41-s + 24·44-s + 490·49-s + 24·59-s + 460·61-s − 64·64-s − 240·71-s + 256·76-s + 1.47e3·79-s − 240·89-s − 66·101-s + 1.11e3·109-s − 408·116-s − 2.63e3·121-s − 664·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 0.164·11-s + 1/4·16-s − 0.772·19-s + 0.653·29-s + 0.961·31-s + 0.822·41-s + 0.0822·44-s + 10/7·49-s + 0.0529·59-s + 0.965·61-s − 1/8·64-s − 0.401·71-s + 0.386·76-s + 2.10·79-s − 0.285·89-s − 0.0650·101-s + 0.977·109-s − 0.326·116-s − 1.97·121-s − 0.480·124-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.153648196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.153648196\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 10 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 2185 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8305 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 32 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 14533 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 51 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 83 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 2710 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 69613 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 74315 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 20342 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 230 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 118 p^{2} T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 445202 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 739 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 35822 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 870818 T^{2} + p^{6} T^{4} \) |
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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
−9.361702076877722449120526590881, −9.124353498034871955933401690878, −8.453453793354451309018099884966, −8.394876211917085261104174862254, −7.903869073135080094233914841176, −7.49016966611487576350335477852, −6.82535018768601007796738271658, −6.73092603130118915292817630716, −6.08010509069653257946722363455, −5.65086271300422371636942208279, −5.34868577788745786674558776190, −4.56058852101003776212100927820, −4.47360631546898143207112590026, −3.95530514552385491804778621995, −3.31460734757970301811369214046, −2.85276309083706174517026952633, −2.24239543669308957958127767314, −1.75116497449538223813241330012, −0.68899607736188490081837866040, −0.64961990104509815895738544281,
0.64961990104509815895738544281, 0.68899607736188490081837866040, 1.75116497449538223813241330012, 2.24239543669308957958127767314, 2.85276309083706174517026952633, 3.31460734757970301811369214046, 3.95530514552385491804778621995, 4.47360631546898143207112590026, 4.56058852101003776212100927820, 5.34868577788745786674558776190, 5.65086271300422371636942208279, 6.08010509069653257946722363455, 6.73092603130118915292817630716, 6.82535018768601007796738271658, 7.49016966611487576350335477852, 7.903869073135080094233914841176, 8.394876211917085261104174862254, 8.453453793354451309018099884966, 9.124353498034871955933401690878, 9.361702076877722449120526590881
