| L(s) = 1 | + 16·4-s − 24·11-s + 192·16-s − 104·23-s + 1.72e3·25-s + 1.40e3·29-s − 3.39e3·37-s − 2.02e3·43-s − 384·44-s + 1.66e4·53-s + 2.04e3·64-s − 2.08e4·67-s + 1.98e3·71-s − 2.96e4·79-s − 1.66e3·92-s + 2.75e4·100-s + 4.07e3·107-s + 7.04e4·109-s − 7.87e4·113-s + 2.25e4·116-s − 1.57e4·121-s + 127-s + 131-s + 137-s + 139-s − 5.42e4·148-s + 149-s + ⋯ |
| L(s) = 1 | + 4-s − 0.198·11-s + 3/4·16-s − 0.196·23-s + 2.75·25-s + 1.67·29-s − 2.47·37-s − 1.09·43-s − 0.198·44-s + 5.93·53-s + 1/2·64-s − 4.63·67-s + 0.393·71-s − 4.74·79-s − 0.196·92-s + 2.75·100-s + 0.355·107-s + 5.92·109-s − 6.16·113-s + 1.67·116-s − 1.07·121-s + 6.20e−5·127-s + 5.82e−5·131-s + 5.32e−5·137-s + 5.17e−5·139-s − 2.47·148-s + 4.50e−5·149-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(5-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{8}\right)^{s/2} \, \Gamma_{\C}(s+2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.6466436648\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6466436648\) |
| \(L(3)\) |
|
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 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 5 | $C_2^2:C_4$ | \( 1 - 1724 T^{2} + 1374142 T^{4} - 1724 p^{8} T^{6} + p^{16} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 + 12 T + 8100 T^{2} + 12 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 13 | $C_2^2:C_4$ | \( 1 - 35804 T^{2} + 1120765054 T^{4} - 35804 p^{8} T^{6} + p^{16} T^{8} \) |
| 17 | $C_2^2:C_4$ | \( 1 + 109328 T^{2} + 11607734400 T^{4} + 109328 p^{8} T^{6} + p^{16} T^{8} \) |
| 19 | $C_2^2:C_4$ | \( 1 - 349488 T^{2} + 60105434976 T^{4} - 349488 p^{8} T^{6} + p^{16} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 + 52 T + 518886 T^{2} + 52 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 - 704 T + 1537498 T^{2} - 704 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 31 | $C_2^2:C_4$ | \( 1 - 1723364 T^{2} + 1763753061958 T^{4} - 1723364 p^{8} T^{6} + p^{16} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 1696 T + 4464834 T^{2} + 1696 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 41 | $C_2^2:C_4$ | \( 1 - 10204640 T^{2} + 41989966346560 T^{4} - 10204640 p^{8} T^{6} + p^{16} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 + 1012 T + 4332388 T^{2} + 1012 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 47 | $C_2^2:C_4$ | \( 1 + 523468 T^{2} - 21953751403130 T^{4} + 523468 p^{8} T^{6} + p^{16} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 8340 T + 32172990 T^{2} - 8340 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 59 | $C_2^2:C_4$ | \( 1 - 9053280 T^{2} + 308187661511040 T^{4} - 9053280 p^{8} T^{6} + p^{16} T^{8} \) |
| 61 | $C_2^2:C_4$ | \( 1 - 45572348 T^{2} + 900315190880446 T^{4} - 45572348 p^{8} T^{6} + p^{16} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 10408 T + 66148266 T^{2} + 10408 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 - 992 T + 47974306 T^{2} - 992 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 73 | $C_2^2:C_4$ | \( 1 - 75977280 T^{2} + 3047966707757760 T^{4} - 75977280 p^{8} T^{6} + p^{16} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 14808 T + 122413578 T^{2} + 14808 p^{4} T^{3} + p^{8} T^{4} )^{2} \) |
| 83 | $C_2^2:C_4$ | \( 1 - 95940864 T^{2} + 6613088999857824 T^{4} - 95940864 p^{8} T^{6} + p^{16} T^{8} \) |
| 89 | $C_2^2:C_4$ | \( 1 - 143773584 T^{2} + 11740110102560544 T^{4} - 143773584 p^{8} T^{6} + p^{16} T^{8} \) |
| 97 | $C_2^2:C_4$ | \( 1 - 240657696 T^{2} + 28374663059580288 T^{4} - 240657696 p^{8} T^{6} + p^{16} T^{8} \) |
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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
−6.73358322424236619299077928405, −6.69238886922514729614678206804, −6.28947206002876355304063726895, −5.91462660514995331153210304986, −5.65933989148956084959664085215, −5.64226410698589984551893332966, −5.41453347474129537843072411074, −4.96324594642556271356790913802, −4.76585624823251908055592094334, −4.76041735526901602208643096786, −4.18715838040334664622517300507, −4.15608105342759867615968428768, −3.64917762434786145287767941705, −3.59610640567540339264320358646, −3.10647737106999495170724098661, −2.94076628793862307764540512706, −2.62697420434530297063870147519, −2.50568495334909828582092148002, −2.36011147685952423187654662312, −1.64127366422150238210963278696, −1.44522177720359180552738795983, −1.34566828847757733343170491413, −0.958869225167929989307498791344, −0.60044295358059914142125294843, −0.07571366206779459053786680440,
0.07571366206779459053786680440, 0.60044295358059914142125294843, 0.958869225167929989307498791344, 1.34566828847757733343170491413, 1.44522177720359180552738795983, 1.64127366422150238210963278696, 2.36011147685952423187654662312, 2.50568495334909828582092148002, 2.62697420434530297063870147519, 2.94076628793862307764540512706, 3.10647737106999495170724098661, 3.59610640567540339264320358646, 3.64917762434786145287767941705, 4.15608105342759867615968428768, 4.18715838040334664622517300507, 4.76041735526901602208643096786, 4.76585624823251908055592094334, 4.96324594642556271356790913802, 5.41453347474129537843072411074, 5.64226410698589984551893332966, 5.65933989148956084959664085215, 5.91462660514995331153210304986, 6.28947206002876355304063726895, 6.69238886922514729614678206804, 6.73358322424236619299077928405
