| L(s) = 1 | + 5·5-s − 17·7-s − 30·11-s + 61·13-s + 240·17-s − 86·19-s + 90·23-s − 90·29-s − 8·31-s − 85·35-s + 634·37-s − 30·41-s + 220·43-s − 180·47-s + 343·49-s + 1.26e3·53-s − 150·55-s − 840·59-s − 599·61-s + 305·65-s − 107·67-s + 420·71-s − 842·73-s + 510·77-s − 353·79-s − 1.35e3·83-s + 1.20e3·85-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.917·7-s − 0.822·11-s + 1.30·13-s + 3.42·17-s − 1.03·19-s + 0.815·23-s − 0.576·29-s − 0.0463·31-s − 0.410·35-s + 2.81·37-s − 0.114·41-s + 0.780·43-s − 0.558·47-s + 49-s + 3.26·53-s − 0.367·55-s − 1.85·59-s − 1.25·61-s + 0.582·65-s − 0.195·67-s + 0.702·71-s − 1.34·73-s + 0.754·77-s − 0.502·79-s − 1.78·83-s + 1.53·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2624400 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.465865207\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.465865207\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 - p T + p^{2} T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - 20 T + p^{3} T^{2} )( 1 + 37 T + p^{3} T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + 30 T - 431 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 61 T + 1524 T^{2} - 61 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 120 T + p^{3} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 43 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 90 T - 4067 T^{2} - 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 90 T - 16289 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 8 T - 29727 T^{2} + 8 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 317 T + p^{3} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 30 T - 68021 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 220 T - 31107 T^{2} - 220 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 180 T - 71423 T^{2} + 180 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 630 T + p^{3} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 840 T + 500221 T^{2} + 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 599 T + 131820 T^{2} + 599 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 107 T - 289314 T^{2} + 107 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 210 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 421 T + p^{3} T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 353 T - 368430 T^{2} + 353 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 1350 T + 1250713 T^{2} + 1350 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 1020 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 997 T + 81336 T^{2} - 997 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.434673149188786150634891185957, −8.655261059702708027989546519692, −8.590935895911703340564045796747, −7.984457040210962597129750903941, −7.66071636604023949444920666462, −7.24614738592119624749236729541, −6.95079950311123835864267019887, −6.03358026341586077025676730573, −6.01860964650562850887763737225, −5.67265966442837578900017165091, −5.45053289540352048151872429596, −4.50588149543507738860548881522, −4.28822408120018169972306177560, −3.52339643377192739457138804914, −3.29268953439903764763847847533, −2.77352787134982938783461526072, −2.37048485838083324022874539221, −1.32572622686098566674194556476, −1.14614287231187465181709167351, −0.45553647347198181822338590956,
0.45553647347198181822338590956, 1.14614287231187465181709167351, 1.32572622686098566674194556476, 2.37048485838083324022874539221, 2.77352787134982938783461526072, 3.29268953439903764763847847533, 3.52339643377192739457138804914, 4.28822408120018169972306177560, 4.50588149543507738860548881522, 5.45053289540352048151872429596, 5.67265966442837578900017165091, 6.01860964650562850887763737225, 6.03358026341586077025676730573, 6.95079950311123835864267019887, 7.24614738592119624749236729541, 7.66071636604023949444920666462, 7.984457040210962597129750903941, 8.590935895911703340564045796747, 8.655261059702708027989546519692, 9.434673149188786150634891185957
