| L(s) = 1 | − 12·3-s + 3·4-s + 90·9-s − 36·12-s − 72·13-s + 41·16-s + 216·17-s + 120·23-s + 228·25-s − 540·27-s − 48·29-s + 270·36-s + 864·39-s + 1.06e3·43-s − 492·48-s + 328·49-s − 2.59e3·51-s − 216·52-s − 864·53-s − 2.28e3·61-s + 411·64-s + 648·68-s − 1.44e3·69-s − 2.73e3·75-s + 288·79-s + 2.83e3·81-s + 576·87-s + ⋯ |
| L(s) = 1 | − 2.30·3-s + 3/8·4-s + 10/3·9-s − 0.866·12-s − 1.53·13-s + 0.640·16-s + 3.08·17-s + 1.08·23-s + 1.82·25-s − 3.84·27-s − 0.307·29-s + 5/4·36-s + 3.54·39-s + 3.77·43-s − 1.47·48-s + 0.956·49-s − 7.11·51-s − 0.576·52-s − 2.23·53-s − 4.78·61-s + 0.802·64-s + 1.15·68-s − 2.51·69-s − 4.21·75-s + 0.410·79-s + 35/9·81-s + 0.709·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2313441 ^{s/2} \, \Gamma_{\C}(s+3/2)^{4} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.014356926\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.014356926\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{4} \) |
| 13 | $D_{4}$ | \( 1 + 72 T + 238 p T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| good | 2 | $D_4\times C_2$ | \( 1 - 3 T^{2} - p^{5} T^{4} - 3 p^{6} T^{6} + p^{12} T^{8} \) |
| 5 | $C_2^2 \wr C_2$ | \( 1 - 228 T^{2} + 33862 T^{4} - 228 p^{6} T^{6} + p^{12} T^{8} \) |
| 7 | $C_2^2 \wr C_2$ | \( 1 - 328 T^{2} + 51918 T^{4} - 328 p^{6} T^{6} + p^{12} T^{8} \) |
| 11 | $C_2^2 \wr C_2$ | \( 1 - 4896 T^{2} + 9533230 T^{4} - 4896 p^{6} T^{6} + p^{12} T^{8} \) |
| 17 | $C_2$ | \( ( 1 - 54 T + p^{3} T^{2} )^{4} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 - 15880 T^{2} + 155242878 T^{4} - 15880 p^{6} T^{6} + p^{12} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 60 T + 1870 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 24 T + 25558 T^{2} + 24 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 102136 T^{2} + 4357504590 T^{4} - 102136 p^{6} T^{6} + p^{12} T^{8} \) |
| 37 | $C_2^2 \wr C_2$ | \( 1 - 2164 T^{2} - 2618990922 T^{4} - 2164 p^{6} T^{6} + p^{12} T^{8} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 230292 T^{2} + 22372016182 T^{4} - 230292 p^{6} T^{6} + p^{12} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 532 T + 206406 T^{2} - 532 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 + 85056 T^{2} + 23167872958 T^{4} + 85056 p^{6} T^{6} + p^{12} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 + 432 T + 134134 T^{2} + 432 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 716592 T^{2} + 212420895502 T^{4} - 716592 p^{6} T^{6} + p^{12} T^{8} \) |
| 61 | $D_{4}$ | \( ( 1 + 1140 T + 12598 p T^{2} + 1140 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 67 | $C_2^2 \wr C_2$ | \( 1 - 475384 T^{2} + 105182398878 T^{4} - 475384 p^{6} T^{6} + p^{12} T^{8} \) |
| 71 | $C_2^2 \wr C_2$ | \( 1 - 1283952 T^{2} + 666179277502 T^{4} - 1283952 p^{6} T^{6} + p^{12} T^{8} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 762916 T^{2} + 359882924838 T^{4} - 762916 p^{6} T^{6} + p^{12} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 144 T + 825118 T^{2} - 144 p^{3} T^{3} + p^{6} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 1633872 T^{2} + 191792062 p^{2} T^{4} - 1633872 p^{6} T^{6} + p^{12} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 2759412 T^{2} + 2896886558902 T^{4} - 2759412 p^{6} T^{6} + p^{12} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 + 53564 T^{2} + 1619815156998 T^{4} + 53564 p^{6} T^{6} + p^{12} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.05709749225251416359284179325, −11.10775630827462632357549398066, −11.01174079882754920464175818625, −10.70823296649040137584100108145, −10.70373797128864100642872704438, −10.22574701848744742654627773305, −9.594210753095342392406161335151, −9.482908196067092387043079038770, −9.469754281697169797448145605460, −8.642078835470311126680523677009, −7.951028238859600759792953353962, −7.68312977319347317698786087600, −7.32923975659664106240264411862, −7.19687916901521482137035701600, −6.70314269080960166989203312143, −6.02624074906807376649485194810, −5.73783415054090777247628409908, −5.70621185884193260495203017744, −4.94541335610298128272522422069, −4.79973349865088929418985526904, −4.26003509010226232746759957385, −3.25494107712853535816675485353, −2.86414057245313290155990252742, −1.38902586585077007976741595606, −0.76420387168859795498402828227,
0.76420387168859795498402828227, 1.38902586585077007976741595606, 2.86414057245313290155990252742, 3.25494107712853535816675485353, 4.26003509010226232746759957385, 4.79973349865088929418985526904, 4.94541335610298128272522422069, 5.70621185884193260495203017744, 5.73783415054090777247628409908, 6.02624074906807376649485194810, 6.70314269080960166989203312143, 7.19687916901521482137035701600, 7.32923975659664106240264411862, 7.68312977319347317698786087600, 7.951028238859600759792953353962, 8.642078835470311126680523677009, 9.469754281697169797448145605460, 9.482908196067092387043079038770, 9.594210753095342392406161335151, 10.22574701848744742654627773305, 10.70373797128864100642872704438, 10.70823296649040137584100108145, 11.01174079882754920464175818625, 11.10775630827462632357549398066, 12.05709749225251416359284179325
