| L(s) = 1 | − 5-s + 2·7-s + 6·13-s − 14·17-s + 14·19-s + 7·23-s + 6·29-s − 3·31-s − 2·35-s − 12·37-s + 4·41-s − 8·43-s − 4·47-s + 7·49-s + 10·53-s + 6·59-s + 3·61-s − 6·65-s + 10·67-s − 24·71-s + 32·73-s − 79-s + 9·83-s + 14·85-s + 8·89-s + 12·91-s − 14·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s + 1.66·13-s − 3.39·17-s + 3.21·19-s + 1.45·23-s + 1.11·29-s − 0.538·31-s − 0.338·35-s − 1.97·37-s + 0.624·41-s − 1.21·43-s − 0.583·47-s + 49-s + 1.37·53-s + 0.781·59-s + 0.384·61-s − 0.744·65-s + 1.22·67-s − 2.84·71-s + 3.74·73-s − 0.112·79-s + 0.987·83-s + 1.51·85-s + 0.847·89-s + 1.25·91-s − 1.43·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10497600 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.372862734\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.372862734\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 - 2 T - 3 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 3 T - 22 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 4 T - 25 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 3 T - 52 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 33 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 16 T + 159 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
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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
−8.801996356893399390472125481729, −8.680432334993304107390338271882, −7.982608841354895772791129993101, −7.919032146873019824566681042626, −7.19359372909318941299191411344, −7.01199020265678373789489845590, −6.58902302956072567093504780022, −6.50811404940060700161877668949, −5.63385297785398887691751685455, −5.35169419095973228186336248913, −5.04968586571594311793446169552, −4.64623014429061818444272699206, −4.17688975278998070546958328272, −3.73783828426390011629091152102, −3.23729461710615140488119303788, −3.02831157640503738755364115502, −2.09706983208405240036698787216, −1.90648445227258967813249004068, −1.01115012071924324615771370582, −0.69713552367859342019892122026,
0.69713552367859342019892122026, 1.01115012071924324615771370582, 1.90648445227258967813249004068, 2.09706983208405240036698787216, 3.02831157640503738755364115502, 3.23729461710615140488119303788, 3.73783828426390011629091152102, 4.17688975278998070546958328272, 4.64623014429061818444272699206, 5.04968586571594311793446169552, 5.35169419095973228186336248913, 5.63385297785398887691751685455, 6.50811404940060700161877668949, 6.58902302956072567093504780022, 7.01199020265678373789489845590, 7.19359372909318941299191411344, 7.919032146873019824566681042626, 7.982608841354895772791129993101, 8.680432334993304107390338271882, 8.801996356893399390472125481729
