| L(s) = 1 | + 4·2-s + 6·3-s + 12·4-s + 10·5-s + 24·6-s + 14·7-s + 32·8-s + 27·9-s + 40·10-s + 32·11-s + 72·12-s − 12·13-s + 56·14-s + 60·15-s + 80·16-s + 4·17-s + 108·18-s + 64·19-s + 120·20-s + 84·21-s + 128·22-s + 32·23-s + 192·24-s + 75·25-s − 48·26-s + 108·27-s + 168·28-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 1.15·3-s + 3/2·4-s + 0.894·5-s + 1.63·6-s + 0.755·7-s + 1.41·8-s + 9-s + 1.26·10-s + 0.877·11-s + 1.73·12-s − 0.256·13-s + 1.06·14-s + 1.03·15-s + 5/4·16-s + 0.0570·17-s + 1.41·18-s + 0.772·19-s + 1.34·20-s + 0.872·21-s + 1.24·22-s + 0.290·23-s + 1.63·24-s + 3/5·25-s − 0.362·26-s + 0.769·27-s + 1.13·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44100 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(13.58125147\) |
| \(L(\frac12)\) |
\(\approx\) |
\(13.58125147\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 3 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 11 | $D_{4}$ | \( 1 - 32 T + 1222 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 12 T - 2354 T^{2} + 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 8134 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 64 T - 522 T^{2} - 64 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 32 T + 22894 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 84 T + 35278 T^{2} + 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 72 T + 54094 T^{2} + 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 60 T + 100510 T^{2} - 60 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 92 T + 31414 T^{2} + 92 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 184 T + 165782 T^{2} + 184 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 208 T + 3746 p T^{2} + 208 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 300 T + 304990 T^{2} + 300 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 568 T + 464278 T^{2} + 568 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 516 T + 478126 T^{2} + 516 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 392 T + 597542 T^{2} - 392 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1448 T + 1212862 T^{2} + 1448 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 756 T + 751318 T^{2} + 756 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 416 T + 1022558 T^{2} - 416 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1752 T + 1666726 T^{2} + 1752 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 956 T + 1529878 T^{2} + 956 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 684 T + 795814 T^{2} - 684 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−12.10004355624751314030076199538, −12.01327589328428280100602298712, −11.22549221486083216235454283312, −10.90409427081517464886079713668, −10.18799236265406109996373276950, −9.712825881143883157610851730718, −9.222028164956310618058083009751, −8.749877742881786164941008582601, −7.983326875348957392153972425694, −7.61711346935075273657007724879, −6.86851435588718998537161364074, −6.60546138407904486784033079150, −5.59268326951291037152293727043, −5.43972795191266605606504529813, −4.37407131971851413854429993266, −4.29121078314455499076657829971, −3.04016591966325908425892249513, −3.00439811988727145033679966412, −1.63352462976968588759969373017, −1.62139934788580866854819176299,
1.62139934788580866854819176299, 1.63352462976968588759969373017, 3.00439811988727145033679966412, 3.04016591966325908425892249513, 4.29121078314455499076657829971, 4.37407131971851413854429993266, 5.43972795191266605606504529813, 5.59268326951291037152293727043, 6.60546138407904486784033079150, 6.86851435588718998537161364074, 7.61711346935075273657007724879, 7.983326875348957392153972425694, 8.749877742881786164941008582601, 9.222028164956310618058083009751, 9.712825881143883157610851730718, 10.18799236265406109996373276950, 10.90409427081517464886079713668, 11.22549221486083216235454283312, 12.01327589328428280100602298712, 12.10004355624751314030076199538
