| L(s) = 1 | + 2-s + 4-s − 5-s + 2·8-s − 9-s − 10-s + 2·16-s − 18-s − 19-s − 20-s + 25-s + 31-s + 2·32-s − 36-s − 38-s − 2·40-s + 2·41-s + 45-s + 2·47-s + 50-s − 59-s + 62-s + 3·64-s − 2·67-s − 2·71-s − 2·72-s − 76-s + ⋯ |
| L(s) = 1 | + 2-s + 4-s − 5-s + 2·8-s − 9-s − 10-s + 2·16-s − 18-s − 19-s − 20-s + 25-s + 31-s + 2·32-s − 36-s − 38-s − 2·40-s + 2·41-s + 45-s + 2·47-s + 50-s − 59-s + 62-s + 3·64-s − 2·67-s − 2·71-s − 2·72-s − 76-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2307361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2307361 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.902428738\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.902428738\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | | \( 1 \) |
| 31 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_1$$\times$$C_2$ | \( ( 1 - T )^{2}( 1 + T + T^{2} ) \) |
| 3 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 5 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 17 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 19 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 43 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 47 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 59 | $C_1$$\times$$C_2$ | \( ( 1 + T )^{2}( 1 - T + T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 83 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 89 | $C_2$ | \( ( 1 - T + T^{2} )( 1 + T + T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
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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
−10.18120124247099726860934638272, −9.356040631438886640778695076989, −8.860705203322919815819196350857, −8.743879257593704398437238735820, −7.944948598441943294811610760986, −7.906186467820005687428357321296, −7.42579268416548263474977017681, −7.14542225264980719802034565323, −6.48425072733216374184969933383, −6.22421965381850730879239178216, −5.68019894342069656689627642347, −5.37159574964039234795200812692, −4.57982136357682714328017470306, −4.41773479427198656877679632591, −4.14707869902630960258815201170, −3.51969767269653018714950428693, −2.78809519777782555222884992936, −2.68790291850424569650232601203, −1.84312791933419256143066304880, −1.00892909238363937758976610376,
1.00892909238363937758976610376, 1.84312791933419256143066304880, 2.68790291850424569650232601203, 2.78809519777782555222884992936, 3.51969767269653018714950428693, 4.14707869902630960258815201170, 4.41773479427198656877679632591, 4.57982136357682714328017470306, 5.37159574964039234795200812692, 5.68019894342069656689627642347, 6.22421965381850730879239178216, 6.48425072733216374184969933383, 7.14542225264980719802034565323, 7.42579268416548263474977017681, 7.906186467820005687428357321296, 7.944948598441943294811610760986, 8.743879257593704398437238735820, 8.860705203322919815819196350857, 9.356040631438886640778695076989, 10.18120124247099726860934638272
