| L(s) = 1 | − 16·2-s + 156·3-s − 870·5-s − 2.49e3·6-s + 4.09e3·8-s + 1.96e4·9-s + 1.39e4·10-s + 5.61e4·11-s + 3.56e5·13-s − 1.35e5·15-s − 6.55e4·16-s + 2.47e5·17-s − 3.14e5·18-s − 3.15e5·19-s − 8.98e5·22-s − 2.04e5·23-s + 6.38e5·24-s + 1.95e6·25-s − 5.69e6·26-s + 5.41e6·27-s − 7.68e6·29-s + 2.17e6·30-s + 1.30e6·31-s + 8.75e6·33-s − 3.96e6·34-s − 4.30e6·37-s + 5.04e6·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.11·3-s − 0.622·5-s − 0.786·6-s + 0.353·8-s + 9-s + 0.440·10-s + 1.15·11-s + 3.45·13-s − 0.692·15-s − 1/4·16-s + 0.719·17-s − 0.707·18-s − 0.555·19-s − 0.817·22-s − 0.152·23-s + 0.393·24-s + 25-s − 2.44·26-s + 1.96·27-s − 2.01·29-s + 0.489·30-s + 0.254·31-s + 1.28·33-s − 0.508·34-s − 0.377·37-s + 0.392·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9604 ^{s/2} \, \Gamma_{\C}(s+9/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(4.282701436\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.282701436\) |
| \(L(\frac{11}{2})\) |
|
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 | $C_2$ | \( 1 + p^{4} T + p^{8} T^{2} \) |
| 7 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 - 52 p T + 517 p^{2} T^{2} - 52 p^{10} T^{3} + p^{18} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 174 p T - 47849 p^{2} T^{2} + 174 p^{10} T^{3} + p^{18} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 56148 T + 794650213 T^{2} - 56148 p^{9} T^{3} + p^{18} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 178094 T + p^{9} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 247662 T - 57251410253 T^{2} - 247662 p^{9} T^{3} + p^{18} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 315380 T - 223223153379 T^{2} + 315380 p^{9} T^{3} + p^{18} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 204504 T - 1759330775447 T^{2} + 204504 p^{9} T^{3} + p^{18} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 3840450 T + p^{9} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 1309408 T - 24725072850207 T^{2} - 1309408 p^{9} T^{3} + p^{18} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4307078 T - 111410818896993 T^{2} + 4307078 p^{9} T^{3} + p^{18} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 1512042 T + p^{9} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 33670604 T + p^{9} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 10581072 T - 1007171388433583 T^{2} - 10581072 p^{9} T^{3} + p^{18} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 16616214 T - 3023665024108337 T^{2} + 16616214 p^{9} T^{3} + p^{18} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 112235100 T + 3933721853355061 T^{2} + 112235100 p^{9} T^{3} + p^{18} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 33197218 T - 10592090809894617 T^{2} - 33197218 p^{9} T^{3} + p^{18} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 121372252 T - 12475310840743443 T^{2} - 121372252 p^{9} T^{3} + p^{18} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 387172728 T + p^{9} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 255240074 T + 6275908667257563 T^{2} + 255240074 p^{9} T^{3} + p^{18} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 492101840 T + 122312624948767281 T^{2} + 492101840 p^{9} T^{3} + p^{18} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 457420236 T + p^{9} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 31809510 T - 349344558781045109 T^{2} - 31809510 p^{9} T^{3} + p^{18} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 673532062 T + p^{9} T^{2} )^{2} \) |
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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
−12.72219134399930243443221946941, −11.62179186030055508278444810192, −11.17893068751046355613096938536, −10.73698087280372397037574736490, −10.25886181074604993496192813146, −9.284543751702870360251442258041, −9.011253380332231363342384726963, −8.502054267361403645442748783866, −8.392459296701280434852345655915, −7.42309330450854800290696113539, −7.11799163947753063938324747222, −6.02021730331340710961275458795, −5.93870765241139562288391077305, −4.26135060459781489082128791657, −4.14079023949632077095225574602, −3.46113802838044526343692029399, −2.88695637041099718782208756214, −1.45286129106508444172001836773, −1.42413899959531227773376229138, −0.65262565622007130634001076399,
0.65262565622007130634001076399, 1.42413899959531227773376229138, 1.45286129106508444172001836773, 2.88695637041099718782208756214, 3.46113802838044526343692029399, 4.14079023949632077095225574602, 4.26135060459781489082128791657, 5.93870765241139562288391077305, 6.02021730331340710961275458795, 7.11799163947753063938324747222, 7.42309330450854800290696113539, 8.392459296701280434852345655915, 8.502054267361403645442748783866, 9.011253380332231363342384726963, 9.284543751702870360251442258041, 10.25886181074604993496192813146, 10.73698087280372397037574736490, 11.17893068751046355613096938536, 11.62179186030055508278444810192, 12.72219134399930243443221946941
