| L(s) = 1 | − 10·5-s − 44·9-s + 4·13-s + 92·17-s + 75·25-s + 232·29-s − 148·37-s − 480·41-s + 440·45-s − 436·49-s − 476·53-s + 1.35e3·61-s − 40·65-s − 1.95e3·73-s + 1.20e3·81-s − 920·85-s − 2.10e3·89-s + 548·97-s − 1.28e3·101-s + 580·109-s − 244·113-s − 176·117-s − 2.02e3·121-s − 500·125-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 1.62·9-s + 0.0853·13-s + 1.31·17-s + 3/5·25-s + 1.48·29-s − 0.657·37-s − 1.82·41-s + 1.45·45-s − 1.27·49-s − 1.23·53-s + 2.84·61-s − 0.0763·65-s − 3.13·73-s + 1.65·81-s − 1.17·85-s − 2.50·89-s + 0.573·97-s − 1.26·101-s + 0.509·109-s − 0.203·113-s − 0.139·117-s − 1.51·121-s − 0.357·125-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1638400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 44 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 436 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 2022 T^{2} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 46 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 - 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10476 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 54742 T^{2} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 p T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 240 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 108604 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 61236 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 238 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 220318 T^{2} + p^{6} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 678 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 601436 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 240582 T^{2} + p^{6} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 978 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 886078 T^{2} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 483084 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1050 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 274 T + p^{3} T^{2} )^{2} \) |
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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
−8.865024397582155884948667792994, −8.558259978918684191814451251226, −8.319220103576333004935578660068, −8.080615308780335237249834979207, −7.47765800920758344060452704806, −7.13221283893254287337527364566, −6.43895243266082401476759860242, −6.40199972551993587911004738282, −5.54245764841009828813822625908, −5.39226870881797375594504846468, −4.91266688433564974900002609175, −4.37137720256668776793920583355, −3.71901149142790443334133210324, −3.34335628134583690205731754600, −2.92188866341407871154536718998, −2.53075107924500005423419044268, −1.53832001176797599123833877326, −1.04571775778523527492318985968, 0, 0,
1.04571775778523527492318985968, 1.53832001176797599123833877326, 2.53075107924500005423419044268, 2.92188866341407871154536718998, 3.34335628134583690205731754600, 3.71901149142790443334133210324, 4.37137720256668776793920583355, 4.91266688433564974900002609175, 5.39226870881797375594504846468, 5.54245764841009828813822625908, 6.40199972551993587911004738282, 6.43895243266082401476759860242, 7.13221283893254287337527364566, 7.47765800920758344060452704806, 8.080615308780335237249834979207, 8.319220103576333004935578660068, 8.558259978918684191814451251226, 8.865024397582155884948667792994
