| L(s) = 1 | − 2·3-s + 7-s + 3·9-s + 8·19-s − 2·21-s − 6·25-s − 4·27-s − 20·29-s − 16·31-s + 12·37-s + 16·47-s + 49-s − 20·53-s − 16·57-s + 24·59-s + 3·63-s + 12·75-s + 5·81-s + 24·83-s + 40·87-s + 32·93-s − 4·109-s − 24·111-s + 36·113-s − 22·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 0.377·7-s + 9-s + 1.83·19-s − 0.436·21-s − 6/5·25-s − 0.769·27-s − 3.71·29-s − 2.87·31-s + 1.97·37-s + 2.33·47-s + 1/7·49-s − 2.74·53-s − 2.11·57-s + 3.12·59-s + 0.377·63-s + 1.38·75-s + 5/9·81-s + 2.63·83-s + 4.28·87-s + 3.31·93-s − 0.383·109-s − 2.27·111-s + 3.38·113-s − 2·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 395136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.018501895\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.018501895\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
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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
−8.859066757414730662597301630531, −7.78412155789454825511415137770, −7.58650222751818669434282574300, −7.51563481619318596865649774866, −6.86425806713158882041882464361, −6.14074423707739109326089273460, −5.62833547116631972624232515311, −5.47137535045602222376296375819, −5.14130610707559547461857964170, −4.16672984732611838189105200979, −3.85282553104851107098279692910, −3.35041169054840837861924307608, −2.14050086106699432947449394920, −1.72065991354884586405445664376, −0.60315625258051052922665001217,
0.60315625258051052922665001217, 1.72065991354884586405445664376, 2.14050086106699432947449394920, 3.35041169054840837861924307608, 3.85282553104851107098279692910, 4.16672984732611838189105200979, 5.14130610707559547461857964170, 5.47137535045602222376296375819, 5.62833547116631972624232515311, 6.14074423707739109326089273460, 6.86425806713158882041882464361, 7.51563481619318596865649774866, 7.58650222751818669434282574300, 7.78412155789454825511415137770, 8.859066757414730662597301630531
