| L(s) = 1 | + 14·3-s + 42·5-s + 98·7-s − 146·9-s + 716·11-s − 714·13-s + 588·15-s − 1.34e3·17-s + 1.94e3·19-s + 1.37e3·21-s + 1.79e3·23-s − 102·25-s − 3.43e3·27-s − 1.20e3·29-s + 6.80e3·31-s + 1.00e4·33-s + 4.11e3·35-s + 1.46e4·37-s − 9.99e3·39-s + 7.89e3·41-s − 524·43-s − 6.13e3·45-s + 1.83e4·47-s + 7.20e3·49-s − 1.88e4·51-s + 4.51e4·53-s + 3.00e4·55-s + ⋯ |
| L(s) = 1 | + 0.898·3-s + 0.751·5-s + 0.755·7-s − 0.600·9-s + 1.78·11-s − 1.17·13-s + 0.674·15-s − 1.12·17-s + 1.23·19-s + 0.678·21-s + 0.706·23-s − 0.0326·25-s − 0.905·27-s − 0.264·29-s + 1.27·31-s + 1.60·33-s + 0.567·35-s + 1.75·37-s − 1.05·39-s + 0.733·41-s − 0.0432·43-s − 0.451·45-s + 1.21·47-s + 3/7·49-s − 1.01·51-s + 2.20·53-s + 1.34·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+5/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(4.886285566\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.886285566\) |
| \(L(\frac{7}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p^{2} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 14 T + 38 p^{2} T^{2} - 14 p^{5} T^{3} + p^{10} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 42 T + 1866 T^{2} - 42 p^{5} T^{3} + p^{10} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 716 T + 412438 T^{2} - 716 p^{5} T^{3} + p^{10} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 714 T + 814258 T^{2} + 714 p^{5} T^{3} + p^{10} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 1344 T + 2728510 T^{2} + 1344 p^{5} T^{3} + p^{10} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 1946 T + 4229670 T^{2} - 1946 p^{5} T^{3} + p^{10} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 1792 T + 11254510 T^{2} - 1792 p^{5} T^{3} + p^{10} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 1200 T - 16154090 T^{2} + 1200 p^{5} T^{3} + p^{10} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 6804 T + 30441118 T^{2} - 6804 p^{5} T^{3} + p^{10} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 14640 T + 187693126 T^{2} - 14640 p^{5} T^{3} + p^{10} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 7896 T + 209593854 T^{2} - 7896 p^{5} T^{3} + p^{10} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 524 T + 242298998 T^{2} + 524 p^{5} T^{3} + p^{10} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 18396 T + 435885630 T^{2} - 18396 p^{5} T^{3} + p^{10} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 45132 T + 1149514990 T^{2} - 45132 p^{5} T^{3} + p^{10} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 22582 T + 1531622854 T^{2} + 22582 p^{5} T^{3} + p^{10} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 52822 T + 2145929546 T^{2} + 52822 p^{5} T^{3} + p^{10} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 9848 T + 2714812022 T^{2} + 9848 p^{5} T^{3} + p^{10} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 840 T + 427300302 T^{2} - 840 p^{5} T^{3} + p^{10} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 122052 T + 7590539974 T^{2} + 122052 p^{5} T^{3} + p^{10} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 31704 T + 6042098590 T^{2} + 31704 p^{5} T^{3} + p^{10} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 36974 T + 4839605030 T^{2} + 36974 p^{5} T^{3} + p^{10} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 210588 T + 21813719542 T^{2} + 210588 p^{5} T^{3} + p^{10} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 44240 T + 2438219582 T^{2} + 44240 p^{5} T^{3} + p^{10} T^{4} \) |
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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
−13.14959210841487839888564418479, −12.35218262813365880855472090892, −11.69929258647649898641338286672, −11.62407458888393029860897269494, −10.91433123454857759476595358907, −10.16420527229933488865533256252, −9.427210758443261068011771889165, −9.302898452331111683910419773758, −8.708974706593855662726643682894, −8.242881140889070867118187242840, −7.21623899251869558610056237794, −7.18602288216079245381770110527, −5.95460756940385018404705333081, −5.78194312700098192308125298950, −4.54677368825664931722378348313, −4.32110097189527161188995267645, −3.02902640735457113282664998902, −2.58308453377345729371073587136, −1.70215073157916865681329125833, −0.818977437847113664353423538590,
0.818977437847113664353423538590, 1.70215073157916865681329125833, 2.58308453377345729371073587136, 3.02902640735457113282664998902, 4.32110097189527161188995267645, 4.54677368825664931722378348313, 5.78194312700098192308125298950, 5.95460756940385018404705333081, 7.18602288216079245381770110527, 7.21623899251869558610056237794, 8.242881140889070867118187242840, 8.708974706593855662726643682894, 9.302898452331111683910419773758, 9.427210758443261068011771889165, 10.16420527229933488865533256252, 10.91433123454857759476595358907, 11.62407458888393029860897269494, 11.69929258647649898641338286672, 12.35218262813365880855472090892, 13.14959210841487839888564418479
