| L(s) = 1 | + 3-s + 3·5-s + 3·9-s + 9·13-s + 3·15-s − 3·17-s + 8·19-s − 9·23-s + 5·25-s + 8·27-s − 15·29-s + 8·31-s + 9·39-s + 15·41-s − 21·43-s + 9·45-s + 3·47-s + 2·49-s − 3·51-s − 3·53-s + 8·57-s − 3·59-s + 7·61-s + 27·65-s − 5·67-s − 9·69-s − 9·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.34·5-s + 9-s + 2.49·13-s + 0.774·15-s − 0.727·17-s + 1.83·19-s − 1.87·23-s + 25-s + 1.53·27-s − 2.78·29-s + 1.43·31-s + 1.44·39-s + 2.34·41-s − 3.20·43-s + 1.34·45-s + 0.437·47-s + 2/7·49-s − 0.420·51-s − 0.412·53-s + 1.05·57-s − 0.390·59-s + 0.896·61-s + 3.34·65-s − 0.610·67-s − 1.08·69-s − 1.06·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1478656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.740863842\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.740863842\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 - 8 T + p T^{2} \) |
| good | 3 | $C_2^2$ | \( 1 - T - 2 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 - 2 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 9 T + 50 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 15 T + 104 T^{2} + 15 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 15 T + 116 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 8 T + p T^{2} )( 1 + 13 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - 3 T + 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T + 56 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 3 T - 50 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 + 9 T + 10 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 7 T - 30 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 154 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 15 T + 164 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 15 T + 172 T^{2} - 15 p T^{3} + p^{2} T^{4} \) |
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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
−9.613737463799651266481014663386, −9.514038535655649652878248809191, −9.428805445326172294936277762038, −8.587057279022980985520003402159, −8.462408401309023849620288562977, −8.032104236143628588569475981047, −7.40107470901311891332661378989, −7.15160090147679139941393715466, −6.47244652223051024373275277617, −6.02990941586390250197137773875, −5.99907616735547383342897559851, −5.39016111202725860819338940540, −4.81410324125552877133106795505, −4.20744429110713390758247401126, −3.75701938541949445237316644520, −3.35553930924184635747774607117, −2.73387895167499935080275026694, −1.90991638080600337907736422450, −1.63786662380192678424973490117, −0.990804334067465316143698404940,
0.990804334067465316143698404940, 1.63786662380192678424973490117, 1.90991638080600337907736422450, 2.73387895167499935080275026694, 3.35553930924184635747774607117, 3.75701938541949445237316644520, 4.20744429110713390758247401126, 4.81410324125552877133106795505, 5.39016111202725860819338940540, 5.99907616735547383342897559851, 6.02990941586390250197137773875, 6.47244652223051024373275277617, 7.15160090147679139941393715466, 7.40107470901311891332661378989, 8.032104236143628588569475981047, 8.462408401309023849620288562977, 8.587057279022980985520003402159, 9.428805445326172294936277762038, 9.514038535655649652878248809191, 9.613737463799651266481014663386
