| L(s) = 1 | − 9·4-s + 100·11-s + 17·16-s + 88·19-s + 100·29-s + 216·31-s + 800·41-s − 900·44-s − 214·49-s − 100·59-s − 1.03e3·61-s + 423·64-s + 1.40e3·71-s − 792·76-s + 1.03e3·79-s + 3.00e3·89-s − 900·101-s − 3.50e3·109-s − 900·116-s + 4.83e3·121-s − 1.94e3·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + ⋯ |
| L(s) = 1 | − 9/8·4-s + 2.74·11-s + 0.265·16-s + 1.06·19-s + 0.640·29-s + 1.25·31-s + 3.04·41-s − 3.08·44-s − 0.623·49-s − 0.220·59-s − 2.17·61-s + 0.826·64-s + 2.34·71-s − 1.19·76-s + 1.46·79-s + 3.57·89-s − 0.886·101-s − 3.08·109-s − 0.720·116-s + 3.63·121-s − 1.40·124-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 + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 50625 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.416463572\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.416463572\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + 9 T^{2} + p^{6} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 214 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 50 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 3994 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9726 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 44 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 9934 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 50 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 108 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 99706 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 400 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 80614 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 129246 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 74346 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 50 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 518 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 569126 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 700 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 609934 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 516 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 707974 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1500 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 831554 T^{2} + p^{6} T^{4} \) |
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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
−12.17164731416797132960391413615, −11.77435758496017140313736896269, −10.94299511453729755118206110675, −10.80871253320518527654999476492, −9.657245757134067183327530444274, −9.541088492914208136454206434538, −9.274144744209553996382140456511, −8.776486951253810735962028177705, −8.145144510953457086409545476407, −7.62866010655286319149762579498, −6.92123420797246701502859318480, −6.18986797736711358031040191399, −6.18243108001068588690407687760, −4.98422031162526763913619135692, −4.64671016246611190302920272824, −3.91793993002735070793711826197, −3.61787867834486799364124222991, −2.54338174539633863164390961164, −1.27265649963728689131021305035, −0.792100579264464299029250631769,
0.792100579264464299029250631769, 1.27265649963728689131021305035, 2.54338174539633863164390961164, 3.61787867834486799364124222991, 3.91793993002735070793711826197, 4.64671016246611190302920272824, 4.98422031162526763913619135692, 6.18243108001068588690407687760, 6.18986797736711358031040191399, 6.92123420797246701502859318480, 7.62866010655286319149762579498, 8.145144510953457086409545476407, 8.776486951253810735962028177705, 9.274144744209553996382140456511, 9.541088492914208136454206434538, 9.657245757134067183327530444274, 10.80871253320518527654999476492, 10.94299511453729755118206110675, 11.77435758496017140313736896269, 12.17164731416797132960391413615
