| L(s) = 1 | + 19·3-s + 131·7-s + 225·9-s − 242·11-s + 1.61e3·13-s + 321·17-s + 3.43e3·19-s + 2.48e3·21-s − 4.53e3·23-s + 6.30e3·27-s − 5.03e3·29-s − 4.46e3·31-s − 4.59e3·33-s + 2.42e3·37-s + 3.05e4·39-s + 1.96e4·41-s + 1.49e4·43-s + 2.30e4·47-s − 9.73e3·49-s + 6.09e3·51-s + 7.20e3·53-s + 6.53e4·57-s + 5.08e4·59-s − 4.41e4·61-s + 2.94e4·63-s + 1.06e4·67-s − 8.61e4·69-s + ⋯ |
| L(s) = 1 | + 1.21·3-s + 1.01·7-s + 0.925·9-s − 0.603·11-s + 2.64·13-s + 0.269·17-s + 2.18·19-s + 1.23·21-s − 1.78·23-s + 1.66·27-s − 1.11·29-s − 0.834·31-s − 0.734·33-s + 0.290·37-s + 3.22·39-s + 1.82·41-s + 1.22·43-s + 1.52·47-s − 0.579·49-s + 0.328·51-s + 0.352·53-s + 2.66·57-s + 1.90·59-s − 1.52·61-s + 0.935·63-s + 0.288·67-s − 2.17·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1210000 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(11.60120995\) |
| \(L(\frac12)\) |
\(\approx\) |
\(11.60120995\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 19 T + 136 T^{2} - 19 p^{5} T^{3} + p^{10} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 131 T + 26898 T^{2} - 131 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 1610 T + 1247970 T^{2} - 1610 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 321 T - 1276838 T^{2} - 321 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 181 p T + 7887306 T^{2} - 181 p^{6} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 4536 T + 14290234 T^{2} + 4536 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5031 T + 38843968 T^{2} + 5031 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 4463 T + 56616342 T^{2} + 4463 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 2423 T + 88962936 T^{2} - 2423 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 19668 T + 301342822 T^{2} - 19668 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 14900 T + 309896886 T^{2} - 14900 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 23052 T + 368356594 T^{2} - 23052 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 7203 T + 763741528 T^{2} - 7203 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 50838 T + 1936923838 T^{2} - 50838 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 44177 T + 2028483204 T^{2} + 44177 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 10610 T + 2706695958 T^{2} - 10610 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1089 T + 3235317082 T^{2} + 1089 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 92654 T + 6263930946 T^{2} - 92654 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 16334 T + 2724732846 T^{2} + 16334 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 88410 T + 7040835670 T^{2} + 88410 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 80817 T + 12691343080 T^{2} + 80817 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 102694 T + 19805064882 T^{2} + 102694 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−9.176894516032484860057997120476, −8.961364980937402005698639265930, −8.307029742220943211507899661529, −8.246946280354957252768157950035, −7.74778122326851753336330315412, −7.51677092283527981618101341444, −7.05942510600590970271029467172, −6.35670316316162626258771306644, −5.74128019242736873154683043532, −5.67445257104952971417818816104, −5.15918070736079742245679086494, −4.30307924368377175311274165247, −4.00770618111618236821536412602, −3.67458486274316711477940281776, −3.12734199964040000042329569679, −2.62303901742709195399658719509, −2.05775317636561767119593363302, −1.47677756649960857181038236173, −1.08351648550918211608498097422, −0.63125785332366393311861942954,
0.63125785332366393311861942954, 1.08351648550918211608498097422, 1.47677756649960857181038236173, 2.05775317636561767119593363302, 2.62303901742709195399658719509, 3.12734199964040000042329569679, 3.67458486274316711477940281776, 4.00770618111618236821536412602, 4.30307924368377175311274165247, 5.15918070736079742245679086494, 5.67445257104952971417818816104, 5.74128019242736873154683043532, 6.35670316316162626258771306644, 7.05942510600590970271029467172, 7.51677092283527981618101341444, 7.74778122326851753336330315412, 8.246946280354957252768157950035, 8.307029742220943211507899661529, 8.961364980937402005698639265930, 9.176894516032484860057997120476
