| L(s) = 1 | − 4·4-s + 16·16-s − 44·29-s − 124·41-s + 82·49-s − 116·61-s − 64·64-s − 284·89-s − 244·101-s − 76·109-s + 176·116-s + 242·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 496·164-s + 167-s − 338·169-s + 173-s + 179-s + 181-s + 191-s + ⋯ |
| L(s) = 1 | − 4-s + 16-s − 1.51·29-s − 3.02·41-s + 1.67·49-s − 1.90·61-s − 64-s − 3.19·89-s − 2.41·101-s − 0.697·109-s + 1.51·116-s + 2·121-s + 0.00787·127-s + 0.00763·131-s + 0.00729·137-s + 0.00719·139-s + 0.00671·149-s + 0.00662·151-s + 0.00636·157-s + 0.00613·163-s + 3.02·164-s + 0.00598·167-s − 2·169-s + 0.00578·173-s + 0.00558·179-s + 0.00552·181-s + 0.00523·191-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6318038038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6318038038\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 82 T^{2} + p^{4} T^{4} \) |
| 11 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 23 | $C_2^2$ | \( 1 + 878 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 22 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 62 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 2078 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 4402 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 58 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 4478 T^{2} + p^{4} T^{4} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8002 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 142 T + p^{2} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.02774104821915182413621120170, −9.577876250556927170591658690110, −9.448083517539989248705096480525, −8.769657172680561004186087582568, −8.507174535346738233606356686592, −8.191167175440060067347233250276, −7.62029537750107254286895566345, −7.07142534181749433030571125740, −6.91893708378497632174186433153, −6.03643660034004665859462621001, −5.77483167076938673251207450493, −5.23283913121832275437019482491, −4.90066039441487419300059224535, −4.21309215128560958340864887298, −3.90978388260583973806455883730, −3.29678485127138449705878856317, −2.80536450053952494591029527338, −1.84764295490495388321207393378, −1.34114430161747284119835119523, −0.27367963529671785378984330852,
0.27367963529671785378984330852, 1.34114430161747284119835119523, 1.84764295490495388321207393378, 2.80536450053952494591029527338, 3.29678485127138449705878856317, 3.90978388260583973806455883730, 4.21309215128560958340864887298, 4.90066039441487419300059224535, 5.23283913121832275437019482491, 5.77483167076938673251207450493, 6.03643660034004665859462621001, 6.91893708378497632174186433153, 7.07142534181749433030571125740, 7.62029537750107254286895566345, 8.191167175440060067347233250276, 8.507174535346738233606356686592, 8.769657172680561004186087582568, 9.448083517539989248705096480525, 9.577876250556927170591658690110, 10.02774104821915182413621120170
