| L(s) = 1 | − 2-s + 3-s + 3·5-s − 6-s − 5·7-s + 8-s − 3·10-s − 11-s − 12·13-s + 5·14-s + 3·15-s − 16-s + 2·17-s + 2·19-s − 5·21-s + 22-s + 23-s + 24-s + 5·25-s + 12·26-s − 27-s − 14·29-s − 3·30-s − 7·31-s − 33-s − 2·34-s − 15·35-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.577·3-s + 1.34·5-s − 0.408·6-s − 1.88·7-s + 0.353·8-s − 0.948·10-s − 0.301·11-s − 3.32·13-s + 1.33·14-s + 0.774·15-s − 1/4·16-s + 0.485·17-s + 0.458·19-s − 1.09·21-s + 0.213·22-s + 0.208·23-s + 0.204·24-s + 25-s + 2.35·26-s − 0.192·27-s − 2.59·29-s − 0.547·30-s − 1.25·31-s − 0.174·33-s − 0.342·34-s − 2.53·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 933156 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3218244612\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3218244612\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | $C_2$ | \( 1 + 5 T + p T^{2} \) |
| 23 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 6 T - 11 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 11 T + 68 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 15 T + 166 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 2 T - 57 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 2 T - 63 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 11 T + 42 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 8 T - 25 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
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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
−10.05589631496281501368192555717, −9.676213546611871581085748678867, −9.487620842230493111988887035186, −9.270344512506887448638965920088, −8.870372306342280414195622581316, −8.150304588394075340463508576525, −7.48257196277459954240382371188, −7.37049190507704357889919229485, −7.14320863162329243498013726139, −6.30777994158210965966384176577, −6.15555771762743253557862360701, −5.34602186115786228687311210255, −5.09353705955192793856188714465, −4.76075201385361770602888148257, −3.50795430069871278115690057824, −3.42561085167064956658946670582, −2.76383080759142530179728279853, −2.04828039071573666403744488370, −1.92237527958281162874417283040, −0.26067682897197828543551700295,
0.26067682897197828543551700295, 1.92237527958281162874417283040, 2.04828039071573666403744488370, 2.76383080759142530179728279853, 3.42561085167064956658946670582, 3.50795430069871278115690057824, 4.76075201385361770602888148257, 5.09353705955192793856188714465, 5.34602186115786228687311210255, 6.15555771762743253557862360701, 6.30777994158210965966384176577, 7.14320863162329243498013726139, 7.37049190507704357889919229485, 7.48257196277459954240382371188, 8.150304588394075340463508576525, 8.870372306342280414195622581316, 9.270344512506887448638965920088, 9.487620842230493111988887035186, 9.676213546611871581085748678867, 10.05589631496281501368192555717
