| L(s) = 1 | − 4·4-s + 120·11-s + 16·16-s + 152·19-s + 12·29-s − 464·31-s − 468·41-s − 480·44-s − 338·49-s + 1.32e3·59-s − 980·61-s − 64·64-s − 240·71-s − 608·76-s − 304·79-s − 1.35e3·89-s − 1.59e3·101-s + 1.94e3·109-s − 48·116-s + 8.13e3·121-s + 1.85e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 3.28·11-s + 1/4·16-s + 1.83·19-s + 0.0768·29-s − 2.68·31-s − 1.78·41-s − 1.64·44-s − 0.985·49-s + 2.91·59-s − 2.05·61-s − 1/8·64-s − 0.401·71-s − 0.917·76-s − 0.432·79-s − 1.61·89-s − 1.57·101-s + 1.70·109-s − 0.0384·116-s + 6.11·121-s + 1.34·124-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
| \(L(2)\) |
\(\approx\) |
\(2.891697116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.891697116\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2^2$ | \( 1 + 338 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 60 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3238 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 8062 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 4 p T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p^{3} T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 232 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 83350 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 234 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10730 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 78046 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 248470 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 660 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 490 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 57818 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 120 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 221518 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 152 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 497158 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 678 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 190 p^{2} T^{2} + p^{6} T^{4} \) |
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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
−11.41634391908890801855662568750, −10.36960486207413218562429654310, −9.635198483667549174820465745977, −9.628846829947841016113485089352, −9.200758631150008014799649441554, −8.629838067292762559393991131701, −8.504650111796543161488609042482, −7.48408825553197541344651142689, −7.24285252541707020506546643841, −6.71176049185541299580077309044, −6.34031533826960366872582671706, −5.56132103344646327153951873948, −5.31839359331918009536537271833, −4.47828883224516753490115609939, −3.94683224588907942385580563854, −3.57013973419933974095619786190, −3.10864917349973933654939251243, −1.61124761801604400487030980137, −1.56047244359457693387529961113, −0.59461831291699184534200897269,
0.59461831291699184534200897269, 1.56047244359457693387529961113, 1.61124761801604400487030980137, 3.10864917349973933654939251243, 3.57013973419933974095619786190, 3.94683224588907942385580563854, 4.47828883224516753490115609939, 5.31839359331918009536537271833, 5.56132103344646327153951873948, 6.34031533826960366872582671706, 6.71176049185541299580077309044, 7.24285252541707020506546643841, 7.48408825553197541344651142689, 8.504650111796543161488609042482, 8.629838067292762559393991131701, 9.200758631150008014799649441554, 9.628846829947841016113485089352, 9.635198483667549174820465745977, 10.36960486207413218562429654310, 11.41634391908890801855662568750
