| L(s) = 1 | − 4·4-s + 60·11-s + 16·16-s − 298·19-s − 468·29-s − 434·31-s + 312·41-s − 240·44-s + 157·49-s − 540·59-s + 550·61-s − 64·64-s − 1.32e3·71-s + 1.19e3·76-s − 1.98e3·79-s − 2.97e3·89-s + 1.58e3·101-s + 110·109-s + 1.87e3·116-s + 38·121-s + 1.73e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 1.64·11-s + 1/4·16-s − 3.59·19-s − 2.99·29-s − 2.51·31-s + 1.18·41-s − 0.822·44-s + 0.457·49-s − 1.19·59-s + 1.15·61-s − 1/8·64-s − 2.20·71-s + 1.79·76-s − 2.82·79-s − 3.54·89-s + 1.56·101-s + 0.0966·109-s + 1.49·116-s + 0.0285·121-s + 1.25·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2230520668\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2230520668\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 157 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 30 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3553 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3742 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 149 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 1834 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 234 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 p T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 79990 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 156 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 28475 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 206746 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 6950 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 270 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 275 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 43283 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 660 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 360718 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 992 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 427858 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1488 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 1723585 T^{2} + p^{6} T^{4} \) |
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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
−11.34049128877074950451369352969, −10.53358960292645537653938043216, −9.989055881716923149986199768111, −9.256158678540510885888404412400, −9.206894157394811621806292764527, −8.588820313287452577230624077447, −8.561703714240199960746630726456, −7.44511951434765535121863509472, −7.38259548458025526323059220671, −6.71120695319031575360012296379, −6.05846461248207200947997781735, −5.88725181840462096222507257120, −5.23278090328273771718161066090, −4.26445250138388733076502887102, −3.95334868207678593643790779802, −3.94788512934183259753906038999, −2.77824565820185130436695104473, −1.78494791557084859252438405434, −1.66419828182520318824609204019, −0.14743981015452121709543997636,
0.14743981015452121709543997636, 1.66419828182520318824609204019, 1.78494791557084859252438405434, 2.77824565820185130436695104473, 3.94788512934183259753906038999, 3.95334868207678593643790779802, 4.26445250138388733076502887102, 5.23278090328273771718161066090, 5.88725181840462096222507257120, 6.05846461248207200947997781735, 6.71120695319031575360012296379, 7.38259548458025526323059220671, 7.44511951434765535121863509472, 8.561703714240199960746630726456, 8.588820313287452577230624077447, 9.206894157394811621806292764527, 9.256158678540510885888404412400, 9.989055881716923149986199768111, 10.53358960292645537653938043216, 11.34049128877074950451369352969
