| L(s) = 1 | − 4·7-s + 4·11-s − 8·23-s − 24·37-s + 12·43-s + 8·49-s + 4·53-s + 4·67-s + 24·71-s − 16·77-s + 17·81-s + 12·107-s + 48·113-s − 34·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 32·161-s + 163-s + 167-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 1.51·7-s + 1.20·11-s − 1.66·23-s − 3.94·37-s + 1.82·43-s + 8/7·49-s + 0.549·53-s + 0.488·67-s + 2.84·71-s − 1.82·77-s + 17/9·81-s + 1.16·107-s + 4.51·113-s − 3.09·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 2.52·161-s + 0.0783·163-s + 0.0773·167-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9443749383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9443749383\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2^2$ | \( 1 + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| good | 3 | $C_2^3$ | \( 1 - 17 T^{4} + p^{4} T^{8} \) |
| 11 | $C_2$ | \( ( 1 - T + p T^{2} )^{4} \) |
| 13 | $C_2^3$ | \( 1 + 103 T^{4} + p^{4} T^{8} \) |
| 17 | $C_2^3$ | \( 1 + 263 T^{4} + p^{4} T^{8} \) |
| 19 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 23 | $C_2^2$ | \( ( 1 + 4 T + 8 T^{2} + 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 29 | $C_2^2$ | \( ( 1 - 49 T^{2} + p^{2} T^{4} )^{2} \) |
| 31 | $C_2^2$ | \( ( 1 - 52 T^{2} + p^{2} T^{4} )^{2} \) |
| 37 | $C_2^2$ | \( ( 1 + 12 T + 72 T^{2} + 12 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $C_2^2$ | \( ( 1 + 8 T^{2} + p^{2} T^{4} )^{2} \) |
| 43 | $C_2^2$ | \( ( 1 - 6 T + 18 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $C_2^3$ | \( 1 - 2017 T^{4} + p^{4} T^{8} \) |
| 53 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 59 | $C_2^2$ | \( ( 1 + 28 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $C_2^2$ | \( ( 1 - 82 T^{2} + p^{2} T^{4} )^{2} \) |
| 67 | $C_2^2$ | \( ( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{4} \) |
| 73 | $C_2^2$ | \( ( 1 + p^{2} T^{4} )^{2} \) |
| 79 | $C_2^2$ | \( ( 1 + 11 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $C_2^3$ | \( 1 + 7538 T^{4} + p^{4} T^{8} \) |
| 89 | $C_2^2$ | \( ( 1 + 138 T^{2} + p^{2} T^{4} )^{2} \) |
| 97 | $C_2^3$ | \( 1 + 16903 T^{4} + p^{4} T^{8} \) |
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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
−7.72870501283941326180974429356, −7.43275282023006510434238821749, −7.34429990007041802233415763209, −6.89928301812361060258880972407, −6.80956830735188032586319270898, −6.51310654736435271553296441340, −6.42672862580717191547418087734, −6.21015018296273942893591595611, −5.75380949491179843479477291476, −5.61972507577436832496684517610, −5.50877598818272985136276309381, −4.97456812635753227234560193581, −4.70202886114552999682199541047, −4.66546324608245500627467459546, −3.94235624711033108667350660203, −3.85688045322793135192724766793, −3.58411013986060227668253818799, −3.48979161712265632384259784247, −3.31094172158265691858067997645, −2.43321540535345335677756445630, −2.39107749742439765669117329982, −2.17170321123647421157774710098, −1.49650498188424158297971662093, −1.11589356614404738373358188247, −0.32324884696239103516372347832,
0.32324884696239103516372347832, 1.11589356614404738373358188247, 1.49650498188424158297971662093, 2.17170321123647421157774710098, 2.39107749742439765669117329982, 2.43321540535345335677756445630, 3.31094172158265691858067997645, 3.48979161712265632384259784247, 3.58411013986060227668253818799, 3.85688045322793135192724766793, 3.94235624711033108667350660203, 4.66546324608245500627467459546, 4.70202886114552999682199541047, 4.97456812635753227234560193581, 5.50877598818272985136276309381, 5.61972507577436832496684517610, 5.75380949491179843479477291476, 6.21015018296273942893591595611, 6.42672862580717191547418087734, 6.51310654736435271553296441340, 6.80956830735188032586319270898, 6.89928301812361060258880972407, 7.34429990007041802233415763209, 7.43275282023006510434238821749, 7.72870501283941326180974429356
