| L(s) = 1 | − 2-s + 2·4-s − 8·5-s + 7-s − 5·8-s + 8·10-s + 4·11-s − 13-s − 14-s + 5·16-s + 6·17-s − 4·19-s − 16·20-s − 4·22-s + 12·23-s + 38·25-s + 26-s + 2·28-s − 2·29-s − 3·31-s − 10·32-s − 6·34-s − 8·35-s − 3·37-s + 4·38-s + 40·40-s + 2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 4-s − 3.57·5-s + 0.377·7-s − 1.76·8-s + 2.52·10-s + 1.20·11-s − 0.277·13-s − 0.267·14-s + 5/4·16-s + 1.45·17-s − 0.917·19-s − 3.57·20-s − 0.852·22-s + 2.50·23-s + 38/5·25-s + 0.196·26-s + 0.377·28-s − 0.371·29-s − 0.538·31-s − 1.76·32-s − 1.02·34-s − 1.35·35-s − 0.493·37-s + 0.648·38-s + 6.32·40-s + 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 321489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4949461522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4949461522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 + T - T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + T - 12 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 3 T - 28 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 2 T - 37 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 6 T - 17 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 7 T - 18 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 4 T - 73 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 9 T - 16 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
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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
−11.05288109472820010693131814896, −10.93772933647595834867251800788, −10.17838508045594631513693907278, −9.508188961198006303690068855136, −8.900077877119511017468084138857, −8.657568276481484185048125538640, −8.452588493066470824775472935302, −7.73616781858480544046532422907, −7.39960014622448238753020063196, −7.27406392210353027661087138961, −6.65985496867972150904102824922, −6.30254567188635709828391854546, −5.23658811508593701327335221441, −4.88486980128595094110576549777, −3.92579464705749345033450504392, −3.89588499383944008974335264691, −3.06823900998149883107946830523, −2.97333866612490550185210634868, −1.36662467734858769962277702162, −0.49456360792895118560005920690,
0.49456360792895118560005920690, 1.36662467734858769962277702162, 2.97333866612490550185210634868, 3.06823900998149883107946830523, 3.89588499383944008974335264691, 3.92579464705749345033450504392, 4.88486980128595094110576549777, 5.23658811508593701327335221441, 6.30254567188635709828391854546, 6.65985496867972150904102824922, 7.27406392210353027661087138961, 7.39960014622448238753020063196, 7.73616781858480544046532422907, 8.452588493066470824775472935302, 8.657568276481484185048125538640, 8.900077877119511017468084138857, 9.508188961198006303690068855136, 10.17838508045594631513693907278, 10.93772933647595834867251800788, 11.05288109472820010693131814896
