| L(s) = 1 | + 4·2-s + 8·4-s − 4·5-s + 32·8-s − 16·10-s + 62·11-s − 124·13-s + 128·16-s + 84·17-s − 100·19-s − 32·20-s + 248·22-s − 42·23-s + 125·25-s − 496·26-s + 20·29-s + 48·31-s + 256·32-s + 336·34-s + 246·37-s − 400·38-s − 128·40-s + 496·41-s + 136·43-s + 496·44-s − 168·46-s + 324·47-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s − 0.357·5-s + 1.41·8-s − 0.505·10-s + 1.69·11-s − 2.64·13-s + 2·16-s + 1.19·17-s − 1.20·19-s − 0.357·20-s + 2.40·22-s − 0.380·23-s + 25-s − 3.74·26-s + 0.128·29-s + 0.278·31-s + 1.41·32-s + 1.69·34-s + 1.09·37-s − 1.70·38-s − 0.505·40-s + 1.88·41-s + 0.482·43-s + 1.69·44-s − 0.538·46-s + 1.00·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 194481 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.544978311\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.544978311\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p^{2} T + p^{3} T^{2} - p^{5} T^{3} + p^{6} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 4 T - 109 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 62 T + 2513 T^{2} - 62 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 62 T + p^{3} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 84 T + 2143 T^{2} - 84 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 100 T + 3141 T^{2} + 100 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 42 T - 10403 T^{2} + 42 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 48 T - 27487 T^{2} - 48 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 246 T + 9863 T^{2} - 246 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 248 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 68 T + p^{3} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 324 T + 1153 T^{2} - 324 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 258 T - 82313 T^{2} - 258 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 120 T - 190979 T^{2} - 120 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 622 T + 159903 T^{2} + 622 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 904 T + 516453 T^{2} + 904 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 678 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 642 T + 23147 T^{2} - 642 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 740 T + 54561 T^{2} + 740 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 468 T + p^{3} T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 200 T - 664969 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 1266 T + p^{3} T^{2} )^{2} \) |
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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
−10.84327885673245217543925897569, −10.73009130820586390128416074635, −10.07815060372108204938249646774, −9.587771310636884164334517806318, −9.337394503780624791968877700012, −8.632785735193207637240522322908, −7.78160001198494247423809153835, −7.77047382940256863652728793137, −7.04261703421583993768362908340, −6.83468802252789932172451463895, −6.05851747561600477167807362356, −5.62841246532407671667947239360, −5.01609353733475628478868443301, −4.43454341049976369631177694198, −4.26674923330580716486259309958, −3.78079238597213965929770116505, −2.83372343395592904218394437471, −2.44434089531320147633408604591, −1.49572059144149677364253525177, −0.68434121406074016123907004078,
0.68434121406074016123907004078, 1.49572059144149677364253525177, 2.44434089531320147633408604591, 2.83372343395592904218394437471, 3.78079238597213965929770116505, 4.26674923330580716486259309958, 4.43454341049976369631177694198, 5.01609353733475628478868443301, 5.62841246532407671667947239360, 6.05851747561600477167807362356, 6.83468802252789932172451463895, 7.04261703421583993768362908340, 7.77047382940256863652728793137, 7.78160001198494247423809153835, 8.632785735193207637240522322908, 9.337394503780624791968877700012, 9.587771310636884164334517806318, 10.07815060372108204938249646774, 10.73009130820586390128416074635, 10.84327885673245217543925897569
