| L(s) = 1 | + 2·3-s − 2·5-s − 7-s + 3·9-s + 11-s − 2·13-s − 4·15-s + 3·17-s + 2·19-s − 2·21-s + 7·23-s + 3·25-s + 4·27-s − 4·29-s − 2·31-s + 2·33-s + 2·35-s + 5·37-s − 4·39-s + 3·41-s + 8·43-s − 6·45-s + 14·47-s − 5·49-s + 6·51-s − 13·53-s − 2·55-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 0.894·5-s − 0.377·7-s + 9-s + 0.301·11-s − 0.554·13-s − 1.03·15-s + 0.727·17-s + 0.458·19-s − 0.436·21-s + 1.45·23-s + 3/5·25-s + 0.769·27-s − 0.742·29-s − 0.359·31-s + 0.348·33-s + 0.338·35-s + 0.821·37-s − 0.640·39-s + 0.468·41-s + 1.21·43-s − 0.894·45-s + 2.04·47-s − 5/7·49-s + 0.840·51-s − 1.78·53-s − 0.269·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9734400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.270022423\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.270022423\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 5 | $C_1$ | \( ( 1 + T )^{2} \) |
| 13 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 7 | $D_{4}$ | \( 1 + T + 6 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - T + 14 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 3 T + 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 2 T + 6 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 7 T + 50 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 31 | $D_{4}$ | \( 1 + 2 T + 30 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 5 T + 72 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 3 T + 76 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 - 14 T + 110 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 13 T + 140 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 + T + 48 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 71 | $D_{4}$ | \( 1 - 17 T + 206 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 5 T - 42 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 - 21 T + 280 T^{2} - 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 11 T + 216 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
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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
−8.869767245263413086887236588381, −8.624584731082655362424883369530, −7.86080329184644779319616684106, −7.81505967036070242013497267022, −7.56604460644555693041563239528, −7.20349931641095654795280588566, −6.60258166799790019844297703096, −6.55583370893388076493927889733, −5.74366971445149472799046192881, −5.47464369187667541921420252408, −4.78101814352305884474049274050, −4.69959567621780839897815104218, −4.00645512161330821042273752053, −3.63273026327086866157222778032, −3.34787750685089773183626831063, −2.99446207764179010787073874349, −2.18550478131614844493224455513, −2.18110608425180761642306814475, −0.923718055399996828974956536915, −0.794754607653661955118629467762,
0.794754607653661955118629467762, 0.923718055399996828974956536915, 2.18110608425180761642306814475, 2.18550478131614844493224455513, 2.99446207764179010787073874349, 3.34787750685089773183626831063, 3.63273026327086866157222778032, 4.00645512161330821042273752053, 4.69959567621780839897815104218, 4.78101814352305884474049274050, 5.47464369187667541921420252408, 5.74366971445149472799046192881, 6.55583370893388076493927889733, 6.60258166799790019844297703096, 7.20349931641095654795280588566, 7.56604460644555693041563239528, 7.81505967036070242013497267022, 7.86080329184644779319616684106, 8.624584731082655362424883369530, 8.869767245263413086887236588381
