| L(s) = 1 | + 2-s − 4-s − 3·8-s + 6·9-s + 13-s − 16-s + 2·17-s + 6·18-s + 25-s + 26-s + 5·32-s + 2·34-s − 6·36-s − 16·37-s − 2·49-s + 50-s − 52-s − 20·53-s + 16·61-s + 7·64-s − 2·68-s − 18·72-s + 16·73-s − 16·74-s + 27·81-s − 4·89-s − 2·98-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 1.06·8-s + 2·9-s + 0.277·13-s − 1/4·16-s + 0.485·17-s + 1.41·18-s + 1/5·25-s + 0.196·26-s + 0.883·32-s + 0.342·34-s − 36-s − 2.63·37-s − 2/7·49-s + 0.141·50-s − 0.138·52-s − 2.74·53-s + 2.04·61-s + 7/8·64-s − 0.242·68-s − 2.12·72-s + 1.87·73-s − 1.85·74-s + 3·81-s − 0.423·89-s − 0.202·98-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1502800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.850348184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.850348184\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + p T^{2} \) |
| 5 | $C_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 13 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 10 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 + 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.956344814679173560130645750704, −7.42098416202486026176332520911, −6.88151949200386059528180448479, −6.64564283838235847728460954301, −6.24354007413923031454928173288, −5.53696407863215736767361553791, −5.13186745572228832228202757437, −4.82683455452161309392174542646, −4.35613449498539556535200530458, −3.77548766902101660819221290155, −3.54396306238639053983018236165, −3.00824810790453349077050332576, −2.01454927911837243427057514499, −1.56445010767842882439297787333, −0.68068940751151498273957179065,
0.68068940751151498273957179065, 1.56445010767842882439297787333, 2.01454927911837243427057514499, 3.00824810790453349077050332576, 3.54396306238639053983018236165, 3.77548766902101660819221290155, 4.35613449498539556535200530458, 4.82683455452161309392174542646, 5.13186745572228832228202757437, 5.53696407863215736767361553791, 6.24354007413923031454928173288, 6.64564283838235847728460954301, 6.88151949200386059528180448479, 7.42098416202486026176332520911, 7.956344814679173560130645750704
