L(s) = 1 | + 2-s − 2·3-s + 4-s − 2·6-s + 8-s − 3·9-s + 2·11-s − 2·12-s + 4·13-s + 16-s − 2·17-s − 3·18-s + 10·19-s + 2·22-s − 12·23-s − 2·24-s + 4·26-s + 14·27-s + 4·29-s + 12·31-s + 32-s − 4·33-s − 2·34-s − 3·36-s + 8·37-s + 10·38-s − 8·39-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.15·3-s + 1/2·4-s − 0.816·6-s + 0.353·8-s − 9-s + 0.603·11-s − 0.577·12-s + 1.10·13-s + 1/4·16-s − 0.485·17-s − 0.707·18-s + 2.29·19-s + 0.426·22-s − 2.50·23-s − 0.408·24-s + 0.784·26-s + 2.69·27-s + 0.742·29-s + 2.15·31-s + 0.176·32-s − 0.696·33-s − 0.342·34-s − 1/2·36-s + 1.31·37-s + 1.62·38-s − 1.28·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.329953899\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.329953899\) |
\(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 5 | | \( 1 \) |
good | 3 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 13 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 17 | $C_2$$\times$$C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 37 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 83 | $C_2$$\times$$C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 15 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 + 2 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−14.8940629672, −14.2754339531, −14.0249630004, −13.7407879771, −13.3181822032, −12.3320214092, −12.1846203232, −11.6436611983, −11.4154247282, −11.1600742237, −10.3542554150, −9.93072373032, −9.37569265672, −8.59823467768, −8.07172921856, −7.75497054910, −6.54551923808, −6.35306147257, −6.01711432754, −5.34945783065, −4.83783296054, −4.05979688076, −3.29699975126, −2.56583690538, −1.05700989626,
1.05700989626, 2.56583690538, 3.29699975126, 4.05979688076, 4.83783296054, 5.34945783065, 6.01711432754, 6.35306147257, 6.54551923808, 7.75497054910, 8.07172921856, 8.59823467768, 9.37569265672, 9.93072373032, 10.3542554150, 11.1600742237, 11.4154247282, 11.6436611983, 12.1846203232, 12.3320214092, 13.3181822032, 13.7407879771, 14.0249630004, 14.2754339531, 14.8940629672