| L(s) = 1 | + 2-s + 3-s + 6-s − 4·7-s − 8-s + 5·11-s + 7·13-s − 4·14-s − 16-s + 2·17-s + 2·19-s − 4·21-s + 5·22-s + 7·23-s − 24-s + 7·26-s − 27-s − 2·29-s − 4·31-s + 5·33-s + 2·34-s + 3·37-s + 2·38-s + 7·39-s − 10·41-s − 4·42-s − 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.408·6-s − 1.51·7-s − 0.353·8-s + 1.50·11-s + 1.94·13-s − 1.06·14-s − 1/4·16-s + 0.485·17-s + 0.458·19-s − 0.872·21-s + 1.06·22-s + 1.45·23-s − 0.204·24-s + 1.37·26-s − 0.192·27-s − 0.371·29-s − 0.718·31-s + 0.870·33-s + 0.342·34-s + 0.493·37-s + 0.324·38-s + 1.12·39-s − 1.56·41-s − 0.617·42-s − 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3802500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.889345692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.889345692\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | | \( 1 \) |
| 13 | $C_2$ | \( 1 - 7 T + p T^{2} \) |
| good | 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 5 T + 14 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 7 T + 26 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 2 T - 25 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 3 T - 28 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 + 10 T + 59 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 4 T - 27 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 11 T + 60 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 4 T - 51 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 10 T + 11 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 13 T + 72 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
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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
−9.401918532880466254749431345746, −8.952866833512395042544117251012, −8.716269876327146720541757139394, −8.211862733462862036368238262011, −8.068567559049680604987020351496, −7.04414441232574313152419709884, −7.02991608384864593459551947942, −6.44571333887087523804366768045, −6.42371898722541337201004964734, −5.73121857453310271069415470411, −5.51242739066755260021550738695, −4.87125336208170969915182398616, −4.37483599139851940468975224515, −3.77238876767105576499012872896, −3.49497992864835392742309373329, −3.14473669500140515567620464528, −3.08377714721138951058421617737, −1.85752019807660781107699387978, −1.46646285865478527487470455551, −0.64265472307152879028379724385,
0.64265472307152879028379724385, 1.46646285865478527487470455551, 1.85752019807660781107699387978, 3.08377714721138951058421617737, 3.14473669500140515567620464528, 3.49497992864835392742309373329, 3.77238876767105576499012872896, 4.37483599139851940468975224515, 4.87125336208170969915182398616, 5.51242739066755260021550738695, 5.73121857453310271069415470411, 6.42371898722541337201004964734, 6.44571333887087523804366768045, 7.02991608384864593459551947942, 7.04414441232574313152419709884, 8.068567559049680604987020351496, 8.211862733462862036368238262011, 8.716269876327146720541757139394, 8.952866833512395042544117251012, 9.401918532880466254749431345746
