| L(s) = 1 | − 32·4-s + 1.08e3·11-s + 768·16-s − 1.18e3·19-s − 1.14e4·29-s − 1.14e4·31-s + 3.08e4·41-s − 3.45e4·44-s − 1.12e4·49-s − 1.01e5·59-s − 5.82e4·61-s − 1.63e4·64-s + 1.23e4·71-s + 3.78e4·76-s − 1.58e4·79-s − 3.29e5·89-s + 1.83e5·101-s − 3.39e5·109-s + 3.64e5·116-s + 3.78e5·121-s + 3.65e5·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 4-s + 2.69·11-s + 3/4·16-s − 0.752·19-s − 2.51·29-s − 2.13·31-s + 2.86·41-s − 2.69·44-s − 0.668·49-s − 3.77·59-s − 2.00·61-s − 1/2·64-s + 0.290·71-s + 0.752·76-s − 0.285·79-s − 4.40·89-s + 1.78·101-s − 2.73·109-s + 2.51·116-s + 2.35·121-s + 2.13·124-s + 5.50e−6·127-s + 5.09e−6·131-s + 4.55e−6·137-s + 4.38e−6·139-s + 3.69e−6·149-s + 3.56e−6·151-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(6-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 5^{8}\right)^{s/2} \, \Gamma_{\C}(s+5/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.4248578293\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4248578293\) |
| \(L(\frac{7}{2})\) |
|
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 | $C_2$ | \( ( 1 + p^{4} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $D_4\times C_2$ | \( 1 + 11230 T^{2} + 229188723 T^{4} + 11230 p^{10} T^{6} + p^{20} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 540 T + 248086 T^{2} - 540 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 795290 T^{2} + 317466366123 T^{4} - 795290 p^{10} T^{6} + p^{20} T^{8} \) |
| 17 | $D_4\times C_2$ | \( 1 + 1787404 T^{2} + 3941614200102 T^{4} + 1787404 p^{10} T^{6} + p^{20} T^{8} \) |
| 19 | $D_{4}$ | \( ( 1 + 592 T + 4121589 T^{2} + 592 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 23 | $D_4\times C_2$ | \( 1 - 11701772 T^{2} + 67885190172294 T^{4} - 11701772 p^{10} T^{6} + p^{20} T^{8} \) |
| 29 | $D_{4}$ | \( ( 1 + 5700 T + 48997882 T^{2} + 5700 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 31 | $D_{4}$ | \( ( 1 + 5708 T + 64485393 T^{2} + 5708 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 37 | $D_4\times C_2$ | \( 1 - 171219020 T^{2} + 14244759565515798 T^{4} - 171219020 p^{10} T^{6} + p^{20} T^{8} \) |
| 41 | $D_{4}$ | \( ( 1 - 15420 T + 258100402 T^{2} - 15420 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 43 | $D_4\times C_2$ | \( 1 - 444579674 T^{2} + 90169634607517467 T^{4} - 444579674 p^{10} T^{6} + p^{20} T^{8} \) |
| 47 | $D_4\times C_2$ | \( 1 - 152380028 T^{2} + 88657259104980294 T^{4} - 152380028 p^{10} T^{6} + p^{20} T^{8} \) |
| 53 | $D_4\times C_2$ | \( 1 - 925638980 T^{2} + 424606861627479798 T^{4} - 925638980 p^{10} T^{6} + p^{20} T^{8} \) |
| 59 | $D_{4}$ | \( ( 1 + 50520 T + 1968600982 T^{2} + 50520 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 61 | $D_{4}$ | \( ( 1 + 29126 T + 1603768671 T^{2} + 29126 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 67 | $D_4\times C_2$ | \( 1 - 656459306 T^{2} + 3750274361339663307 T^{4} - 656459306 p^{10} T^{6} + p^{20} T^{8} \) |
| 71 | $D_{4}$ | \( ( 1 - 6180 T + 3034897198 T^{2} - 6180 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 73 | $D_4\times C_2$ | \( 1 - 6693286172 T^{2} + 19222787553262718694 T^{4} - 6693286172 p^{10} T^{6} + p^{20} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 + 7912 T + 1409684334 T^{2} + 7912 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 83 | $D_4\times C_2$ | \( 1 - 1661342924 T^{2} + 21907932611565084342 T^{4} - 1661342924 p^{10} T^{6} + p^{20} T^{8} \) |
| 89 | $D_{4}$ | \( ( 1 + 164640 T + 13798144114 T^{2} + 164640 p^{5} T^{3} + p^{10} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 23242017410 T^{2} + \)\(26\!\cdots\!23\)\( T^{4} - 23242017410 p^{10} T^{6} + p^{20} T^{8} \) |
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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
−7.25359417144321640556116154801, −7.15643825132269551579814400222, −6.50619841728221176377765294045, −6.48290332109705766062280523027, −6.05101155901543030112681657840, −6.03645951779346264929586872493, −5.68879894281731023273189515588, −5.52969943641685806212438393332, −5.23371439919100636162376420660, −4.61730439561757659816290346270, −4.58780957905680133923913664741, −4.31239713376191532820559607211, −4.15333923998169789058613362733, −3.71382687121261508912707940402, −3.67555807958419296563889101152, −3.37338813094663318505183797089, −3.02451473746637704557482845324, −2.54351808588769703834920611115, −2.24231753754783027460535899930, −1.64122946664554805181071826434, −1.61597752402863575490185789541, −1.23299644825956744436563576699, −1.15178191451035399823532164686, −0.34823503427311567116341680661, −0.11960892290186213402406381799,
0.11960892290186213402406381799, 0.34823503427311567116341680661, 1.15178191451035399823532164686, 1.23299644825956744436563576699, 1.61597752402863575490185789541, 1.64122946664554805181071826434, 2.24231753754783027460535899930, 2.54351808588769703834920611115, 3.02451473746637704557482845324, 3.37338813094663318505183797089, 3.67555807958419296563889101152, 3.71382687121261508912707940402, 4.15333923998169789058613362733, 4.31239713376191532820559607211, 4.58780957905680133923913664741, 4.61730439561757659816290346270, 5.23371439919100636162376420660, 5.52969943641685806212438393332, 5.68879894281731023273189515588, 6.03645951779346264929586872493, 6.05101155901543030112681657840, 6.48290332109705766062280523027, 6.50619841728221176377765294045, 7.15643825132269551579814400222, 7.25359417144321640556116154801
