| L(s) = 1 | + 2·2-s + 3·4-s + 4·8-s − 9-s + 5·16-s + 2·17-s − 2·18-s + 16·19-s + 6·32-s + 4·34-s − 3·36-s + 32·38-s − 12·43-s + 16·47-s + 14·49-s + 12·53-s + 12·59-s + 7·64-s + 20·67-s + 6·68-s − 4·72-s + 48·76-s + 81-s − 24·83-s − 24·86-s + 28·89-s + 32·94-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s + 1.41·8-s − 1/3·9-s + 5/4·16-s + 0.485·17-s − 0.471·18-s + 3.67·19-s + 1.06·32-s + 0.685·34-s − 1/2·36-s + 5.19·38-s − 1.82·43-s + 2.33·47-s + 2·49-s + 1.64·53-s + 1.56·59-s + 7/8·64-s + 2.44·67-s + 0.727·68-s − 0.471·72-s + 5.50·76-s + 1/9·81-s − 2.63·83-s − 2.58·86-s + 2.96·89-s + 3.30·94-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6502500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.091191065\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.091191065\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 58 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - 78 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 - 118 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 126 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 142 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.927739813581548502727448889036, −8.829582464667865790753115584922, −8.210606208758927023307772424561, −7.69024635663303129893962734955, −7.38058368023213306658432373588, −7.27519076508085705940788605567, −6.74491323283084778231222936881, −6.33840774044578753550369545750, −5.70742318080407119775402829541, −5.43078525938185191975391449588, −5.23188865081093386455265389758, −5.07024663955058069175625741776, −4.04488169132531282182022839774, −4.00341824050741123512337264317, −3.41445625715847626277286752721, −3.10937172451887155454848204274, −2.51916764976005732148502215439, −2.20425950977650404109699699955, −1.08794181535961696578815513966, −0.993374860608457341298170477443,
0.993374860608457341298170477443, 1.08794181535961696578815513966, 2.20425950977650404109699699955, 2.51916764976005732148502215439, 3.10937172451887155454848204274, 3.41445625715847626277286752721, 4.00341824050741123512337264317, 4.04488169132531282182022839774, 5.07024663955058069175625741776, 5.23188865081093386455265389758, 5.43078525938185191975391449588, 5.70742318080407119775402829541, 6.33840774044578753550369545750, 6.74491323283084778231222936881, 7.27519076508085705940788605567, 7.38058368023213306658432373588, 7.69024635663303129893962734955, 8.210606208758927023307772424561, 8.829582464667865790753115584922, 8.927739813581548502727448889036
