L(s) = 1 | + 2·5-s − 6·11-s + 12·13-s − 2·17-s + 4·19-s − 2·23-s + 5·25-s + 16·29-s + 4·31-s + 6·37-s − 20·41-s − 8·43-s − 4·47-s + 4·53-s − 12·55-s − 12·59-s − 2·61-s + 24·65-s − 12·67-s + 12·71-s − 2·73-s + 8·79-s − 4·85-s − 14·89-s + 8·95-s + 4·97-s − 18·101-s + ⋯ |
L(s) = 1 | + 0.894·5-s − 1.80·11-s + 3.32·13-s − 0.485·17-s + 0.917·19-s − 0.417·23-s + 25-s + 2.97·29-s + 0.718·31-s + 0.986·37-s − 3.12·41-s − 1.21·43-s − 0.583·47-s + 0.549·53-s − 1.61·55-s − 1.56·59-s − 0.256·61-s + 2.97·65-s − 1.46·67-s + 1.42·71-s − 0.234·73-s + 0.900·79-s − 0.433·85-s − 1.48·89-s + 0.820·95-s + 0.406·97-s − 1.79·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12446784 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.592680089\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.592680089\) |
\(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | | \( 1 \) |
good | 5 | $C_2^2$ | \( 1 - 2 T - T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2^2$ | \( 1 - 4 T - 3 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 2 T - 19 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 37 | $C_2^2$ | \( 1 - 6 T - T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 4 T - 31 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 12 T + 85 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 2 T - 57 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T + 107 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
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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
−8.548731714191857797173996603179, −8.489205427224337275673642677005, −8.181642734127203393826297818771, −7.86569791232421248871365970178, −7.20168697974625027682281684794, −6.68450368100258285853207599550, −6.51724188660796998538176074111, −6.05868621743402411021196542835, −5.97940996027913103930539336202, −5.32127685999385607990761362779, −4.97156680894205134324518776623, −4.72620104924441527983848819359, −4.18001771576302534662991864852, −3.47804354098562059591615173365, −3.18502558688775385687432950433, −2.90088395096359887441530857054, −2.33101633129175884736997843814, −1.48660389731909005918055042225, −1.38605028238985385935584744063, −0.59428116124991897678182422831,
0.59428116124991897678182422831, 1.38605028238985385935584744063, 1.48660389731909005918055042225, 2.33101633129175884736997843814, 2.90088395096359887441530857054, 3.18502558688775385687432950433, 3.47804354098562059591615173365, 4.18001771576302534662991864852, 4.72620104924441527983848819359, 4.97156680894205134324518776623, 5.32127685999385607990761362779, 5.97940996027913103930539336202, 6.05868621743402411021196542835, 6.51724188660796998538176074111, 6.68450368100258285853207599550, 7.20168697974625027682281684794, 7.86569791232421248871365970178, 8.181642734127203393826297818771, 8.489205427224337275673642677005, 8.548731714191857797173996603179