L(s) = 1 | + 2-s − 4-s + 4·7-s − 3·8-s + 4·9-s + 4·14-s − 16-s + 6·17-s + 4·18-s − 2·23-s + 25-s − 4·28-s − 6·31-s + 5·32-s + 6·34-s − 4·36-s − 12·41-s − 2·46-s + 9·49-s + 50-s − 12·56-s − 6·62-s + 16·63-s + 7·64-s − 6·68-s + 2·71-s − 12·72-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1/2·4-s + 1.51·7-s − 1.06·8-s + 4/3·9-s + 1.06·14-s − 1/4·16-s + 1.45·17-s + 0.942·18-s − 0.417·23-s + 1/5·25-s − 0.755·28-s − 1.07·31-s + 0.883·32-s + 1.02·34-s − 2/3·36-s − 1.87·41-s − 0.294·46-s + 9/7·49-s + 0.141·50-s − 1.60·56-s − 0.762·62-s + 2.01·63-s + 7/8·64-s − 0.727·68-s + 0.237·71-s − 1.41·72-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 78400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.286195457\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.286195457\) |
\(L(\frac{3}{2})\) |
|
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$ | \( ( 1 - T )( 1 + T ) \) |
| 7 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
good | 3 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 19 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 6 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 36 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 79 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 112 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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
−9.722541549765708687556305540986, −9.336332197107691456540394854631, −8.694640804898336414451891145456, −8.142851944271096050370078380170, −7.83012183520527868723693039092, −7.23759366639998798389118583271, −6.68915695838909050817435333113, −5.90929681407231256552382587887, −5.27771126255540238709098197802, −5.01623728842849738389832798117, −4.39659444191709495610921940557, −3.82163045100574741785866851101, −3.26838998059552631489767379022, −2.03893946251772486336609709648, −1.24219546809533598465917896886,
1.24219546809533598465917896886, 2.03893946251772486336609709648, 3.26838998059552631489767379022, 3.82163045100574741785866851101, 4.39659444191709495610921940557, 5.01623728842849738389832798117, 5.27771126255540238709098197802, 5.90929681407231256552382587887, 6.68915695838909050817435333113, 7.23759366639998798389118583271, 7.83012183520527868723693039092, 8.142851944271096050370078380170, 8.694640804898336414451891145456, 9.336332197107691456540394854631, 9.722541549765708687556305540986