| L(s) = 1 | − 64·4-s − 729·9-s + 2.36e3·11-s + 4.09e3·16-s − 2.55e4·19-s − 2.16e5·29-s + 2.84e5·31-s + 4.66e4·36-s + 1.05e6·41-s − 1.51e5·44-s + 1.52e6·49-s + 3.16e6·59-s − 1.86e6·61-s − 2.62e5·64-s + 5.92e6·71-s + 1.63e6·76-s + 1.12e7·79-s + 5.31e5·81-s + 1.83e7·89-s − 1.72e6·99-s + 8.59e6·101-s + 1.17e7·109-s + 1.38e7·116-s − 3.47e7·121-s − 1.82e7·124-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1/3·9-s + 0.535·11-s + 1/4·16-s − 0.855·19-s − 1.64·29-s + 1.71·31-s + 1/6·36-s + 2.37·41-s − 0.267·44-s + 1.85·49-s + 2.00·59-s − 1.05·61-s − 1/8·64-s + 1.96·71-s + 0.427·76-s + 2.57·79-s + 1/9·81-s + 2.75·89-s − 0.178·99-s + 0.829·101-s + 0.866·109-s + 0.822·116-s − 1.78·121-s − 0.858·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 22500 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.526656814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.526656814\) |
| \(L(\frac{9}{2})\) |
|
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 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 3 | $C_2$ | \( 1 + p^{6} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 1525285 T^{2} + p^{14} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 1182 T + p^{7} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 122528305 T^{2} + p^{14} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 764517310 T^{2} + p^{14} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 12785 T + p^{7} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 6768665290 T^{2} + p^{14} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 108090 T + p^{7} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 142427 T + p^{7} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 113540851510 T^{2} + p^{14} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 525072 T + p^{7} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 14391199955 T^{2} + p^{14} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 686832842650 T^{2} + p^{14} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 182555846890 T^{2} + p^{14} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 1582110 T + p^{7} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 932893 T + p^{7} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 9271778738725 T^{2} + p^{14} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 2962752 T + p^{7} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 5458203913390 T^{2} + p^{14} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5635360 T + p^{7} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 44535717558130 T^{2} + p^{14} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 9155040 T + p^{7} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 60770891598625 T^{2} + p^{14} T^{4} \) |
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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
−11.82636828319301438400434164846, −11.57506591811237089939578421857, −10.79122301272476736973853167698, −10.56663181391608315598461646475, −9.815237818699109771075795506846, −9.256550778624938870796122984736, −8.987601046980126806472158091204, −8.356525645223756400812148652118, −7.75268929603098850649911124408, −7.33499212566510346807556147191, −6.29464298274463461613977866463, −6.25285436653446412581779815300, −5.33036256236209781946923307669, −4.81552359821568834421920703204, −3.87533583452047023180854267043, −3.81464016843172065053585178199, −2.57863561525709095078163004519, −2.14197945934915213681505355756, −0.988661158234050780755091241704, −0.53497479988527517587139558538,
0.53497479988527517587139558538, 0.988661158234050780755091241704, 2.14197945934915213681505355756, 2.57863561525709095078163004519, 3.81464016843172065053585178199, 3.87533583452047023180854267043, 4.81552359821568834421920703204, 5.33036256236209781946923307669, 6.25285436653446412581779815300, 6.29464298274463461613977866463, 7.33499212566510346807556147191, 7.75268929603098850649911124408, 8.356525645223756400812148652118, 8.987601046980126806472158091204, 9.256550778624938870796122984736, 9.815237818699109771075795506846, 10.56663181391608315598461646475, 10.79122301272476736973853167698, 11.57506591811237089939578421857, 11.82636828319301438400434164846
