| L(s) = 1 | − 9·9-s − 48·11-s − 40·19-s − 612·29-s − 272·31-s − 300·41-s − 98·49-s + 1.48e3·59-s − 836·61-s + 960·71-s − 2.70e3·79-s + 81·81-s + 60·89-s + 432·99-s − 3.08e3·101-s + 3.71e3·109-s − 934·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 506·169-s + ⋯ |
| L(s) = 1 | − 1/3·9-s − 1.31·11-s − 0.482·19-s − 3.91·29-s − 1.57·31-s − 1.14·41-s − 2/7·49-s + 3.28·59-s − 1.75·61-s + 1.60·71-s − 3.85·79-s + 1/9·81-s + 0.0714·89-s + 0.438·99-s − 3.03·101-s + 3.26·109-s − 0.701·121-s + 0.000698·127-s + 0.000666·131-s + 0.000623·137-s + 0.000610·139-s + 0.000549·149-s + 0.000538·151-s + 0.000508·157-s + 0.000480·163-s + 0.000463·167-s − 0.230·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3804612551\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3804612551\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 24 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 + 506 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 p^{2} T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 20 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 19150 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 306 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 136 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 396 T + p^{3} T^{2} )( 1 + 396 T + p^{3} T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 150 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 73750 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 202462 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 126358 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 744 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 418 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 566182 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 480 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 589678 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 1352 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 769030 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1743550 T^{2} + p^{6} T^{4} \) |
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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
−11.34064297322135270274776709491, −11.08878124790016621636380164670, −10.79881034349280661534942936398, −9.931593722838556417466438636177, −9.904395766119424214776347652034, −9.014578247177705852645739610895, −8.870127374877108429017804567616, −8.067908766606554171168212463315, −7.79503360970758461673710478982, −7.07107147013258066738460917740, −6.94549049681046813404907980585, −5.69885639492656934443017549807, −5.68441340119835387532225692903, −5.21092384867302336260013302716, −4.29532029313424573080235283742, −3.70522203594383587188523298040, −3.12361369486874690740920191813, −2.19196243165657908107263875991, −1.73700742106261002077288777173, −0.21581774694381423481913778020,
0.21581774694381423481913778020, 1.73700742106261002077288777173, 2.19196243165657908107263875991, 3.12361369486874690740920191813, 3.70522203594383587188523298040, 4.29532029313424573080235283742, 5.21092384867302336260013302716, 5.68441340119835387532225692903, 5.69885639492656934443017549807, 6.94549049681046813404907980585, 7.07107147013258066738460917740, 7.79503360970758461673710478982, 8.067908766606554171168212463315, 8.870127374877108429017804567616, 9.014578247177705852645739610895, 9.904395766119424214776347652034, 9.931593722838556417466438636177, 10.79881034349280661534942936398, 11.08878124790016621636380164670, 11.34064297322135270274776709491
