| L(s) = 1 | + 2-s + 2·5-s + 4·7-s − 8-s + 2·10-s − 11-s + 6·13-s + 4·14-s − 16-s − 4·17-s + 8·19-s − 22-s + 4·23-s + 5·25-s + 6·26-s + 6·29-s − 4·34-s + 8·35-s + 12·37-s + 8·38-s − 2·40-s − 6·41-s − 4·43-s + 4·46-s − 12·47-s + 7·49-s + 5·50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.894·5-s + 1.51·7-s − 0.353·8-s + 0.632·10-s − 0.301·11-s + 1.66·13-s + 1.06·14-s − 1/4·16-s − 0.970·17-s + 1.83·19-s − 0.213·22-s + 0.834·23-s + 25-s + 1.17·26-s + 1.11·29-s − 0.685·34-s + 1.35·35-s + 1.97·37-s + 1.29·38-s − 0.316·40-s − 0.937·41-s − 0.609·43-s + 0.589·46-s − 1.75·47-s + 49-s + 0.707·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3175524 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.418530687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.418530687\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | | \( 1 \) |
| 11 | $C_2$ | \( 1 + T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 13 | $C_2^2$ | \( 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 6 T + 7 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 6 T - 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 12 T + 85 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 - 4 T - 67 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 19 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
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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
−9.540772325836092351020974804585, −9.034384297390905400548348403705, −8.451011071456022712355878435702, −8.436669977108433953724798262909, −8.121163204432596674383784298862, −7.48494936392234389853953846666, −6.86869760994811571956819844769, −6.77560937620792065712276556211, −6.04873126165091004815406788179, −5.91866392055054951106521898505, −5.23895898386197187467614266012, −4.95941123889200806256257287817, −4.79195880012068590063256394188, −4.21339163362861451125180654096, −3.48279463493006215012458586322, −3.27992994922626912873065124973, −2.50452863707612597124360953207, −2.09958734761892235686198230314, −1.15127893624843364133223562662, −1.10490908248658394564698272641,
1.10490908248658394564698272641, 1.15127893624843364133223562662, 2.09958734761892235686198230314, 2.50452863707612597124360953207, 3.27992994922626912873065124973, 3.48279463493006215012458586322, 4.21339163362861451125180654096, 4.79195880012068590063256394188, 4.95941123889200806256257287817, 5.23895898386197187467614266012, 5.91866392055054951106521898505, 6.04873126165091004815406788179, 6.77560937620792065712276556211, 6.86869760994811571956819844769, 7.48494936392234389853953846666, 8.121163204432596674383784298862, 8.436669977108433953724798262909, 8.451011071456022712355878435702, 9.034384297390905400548348403705, 9.540772325836092351020974804585
