L(s) = 1 | − 4·4-s + 50·9-s − 54·11-s + 16·16-s − 124·19-s − 282·29-s − 248·31-s − 200·36-s − 708·41-s + 216·44-s − 49·49-s − 696·59-s + 820·61-s − 64·64-s − 678·71-s + 496·76-s − 1.46e3·79-s + 1.77e3·81-s − 1.92e3·89-s − 2.70e3·99-s + 2.94e3·101-s + 998·109-s + 1.12e3·116-s − 475·121-s + 992·124-s + 127-s + 131-s + ⋯ |
L(s) = 1 | − 1/2·4-s + 1.85·9-s − 1.48·11-s + 1/4·16-s − 1.49·19-s − 1.80·29-s − 1.43·31-s − 0.925·36-s − 2.69·41-s + 0.740·44-s − 1/7·49-s − 1.53·59-s + 1.72·61-s − 1/8·64-s − 1.13·71-s + 0.748·76-s − 2.08·79-s + 2.42·81-s − 2.28·89-s − 2.74·99-s + 2.89·101-s + 0.876·109-s + 0.902·116-s − 0.356·121-s + 0.718·124-s + 0.000698·127-s + 0.000666·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.7372200973\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7372200973\) |
\(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + p^{2} T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 50 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 27 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 298 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9250 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 13309 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 141 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 4 p T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 91415 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 354 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 114493 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 197242 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 204118 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 348 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 410 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 479725 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 339 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 773134 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 731 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 864790 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 960 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 29746 T^{2} + p^{6} T^{4} \) |
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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
−11.52965996470426884705084849394, −10.54482361091870062466469265850, −10.32875759299709447277402045233, −10.10892822430006430155690565664, −9.497689087696767393107919467453, −8.940200409973147858371027807964, −8.577124298444843599746367814961, −7.82892198967473867272153589300, −7.65176052445653907509547879938, −6.92095039226233986386859747958, −6.75730210864393543489179777664, −5.72890474464308201113530567448, −5.43497261635893412778532237436, −4.72818675987604307057435996847, −4.31131491497541652775909266376, −3.73958255647490258788318272966, −3.06716999397876975228221872792, −1.90859871342238872511482614946, −1.71622393891674759122324223518, −0.29432559864567843462942388783,
0.29432559864567843462942388783, 1.71622393891674759122324223518, 1.90859871342238872511482614946, 3.06716999397876975228221872792, 3.73958255647490258788318272966, 4.31131491497541652775909266376, 4.72818675987604307057435996847, 5.43497261635893412778532237436, 5.72890474464308201113530567448, 6.75730210864393543489179777664, 6.92095039226233986386859747958, 7.65176052445653907509547879938, 7.82892198967473867272153589300, 8.577124298444843599746367814961, 8.940200409973147858371027807964, 9.497689087696767393107919467453, 10.10892822430006430155690565664, 10.32875759299709447277402045233, 10.54482361091870062466469265850, 11.52965996470426884705084849394