| L(s) = 1 | + 6·3-s + 28·7-s + 27·9-s − 104·19-s + 168·21-s + 58·25-s + 108·27-s + 492·29-s + 232·31-s + 628·37-s − 384·47-s + 441·49-s + 300·53-s − 624·57-s + 408·59-s + 756·63-s + 348·75-s + 405·81-s + 504·83-s + 2.95e3·87-s + 1.39e3·93-s − 1.01e3·103-s − 1.94e3·109-s + 3.76e3·111-s − 492·113-s + 2.65e3·121-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.51·7-s + 9-s − 1.25·19-s + 1.74·21-s + 0.463·25-s + 0.769·27-s + 3.15·29-s + 1.34·31-s + 2.79·37-s − 1.19·47-s + 9/7·49-s + 0.777·53-s − 1.45·57-s + 0.900·59-s + 1.51·63-s + 0.535·75-s + 5/9·81-s + 0.666·83-s + 3.63·87-s + 1.55·93-s − 0.971·103-s − 1.71·109-s + 3.22·111-s − 0.409·113-s + 1.99·121-s + 0.000698·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1806336 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(9.591026067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.591026067\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_2$ | \( 1 - 4 p T + p^{3} T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 58 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2650 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 4202 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 4018 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 52 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 11266 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 246 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 116 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 314 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 64834 T^{2} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 16442 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 192 T + p^{3} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 150 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 204 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 115274 T^{2} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 342218 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 53122 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 762482 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 905230 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 252 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 1363810 T^{2} + p^{6} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 251326 T^{2} + p^{6} T^{4} \) |
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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
−9.435933731420810365812391764027, −8.825182923845168660962905998087, −8.395926179581432040461369469369, −8.372469770283260450172791875954, −8.009965443519900090799310916722, −7.67117527291064216357880532105, −6.99161717872348489286416765777, −6.63495826502382250587761592152, −6.29455299096144887572800920874, −5.73493218790902168977932490039, −4.95421450999669982829128978601, −4.71539319953946292687860823571, −4.27134229338648933868876605837, −4.11111524396321962121515738059, −3.08239546601472693535658463718, −2.80333992115501500798035612321, −2.31756275292643405135887190997, −1.82037552739781522090092323424, −0.995458627847810515670471980767, −0.800045534854660215341260278726,
0.800045534854660215341260278726, 0.995458627847810515670471980767, 1.82037552739781522090092323424, 2.31756275292643405135887190997, 2.80333992115501500798035612321, 3.08239546601472693535658463718, 4.11111524396321962121515738059, 4.27134229338648933868876605837, 4.71539319953946292687860823571, 4.95421450999669982829128978601, 5.73493218790902168977932490039, 6.29455299096144887572800920874, 6.63495826502382250587761592152, 6.99161717872348489286416765777, 7.67117527291064216357880532105, 8.009965443519900090799310916722, 8.372469770283260450172791875954, 8.395926179581432040461369469369, 8.825182923845168660962905998087, 9.435933731420810365812391764027
