| L(s) = 1 | − 2·3-s − 4·7-s + 3·9-s − 8·19-s + 8·21-s + 10·25-s − 4·27-s − 12·29-s + 8·31-s − 4·37-s + 9·49-s − 12·53-s + 16·57-s + 24·59-s − 12·63-s − 20·75-s + 5·81-s + 24·83-s + 24·87-s − 16·93-s + 8·103-s − 20·109-s + 8·111-s − 12·113-s + 10·121-s + 127-s + 131-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 1.51·7-s + 9-s − 1.83·19-s + 1.74·21-s + 2·25-s − 0.769·27-s − 2.22·29-s + 1.43·31-s − 0.657·37-s + 9/7·49-s − 1.64·53-s + 2.11·57-s + 3.12·59-s − 1.51·63-s − 2.30·75-s + 5/9·81-s + 2.63·83-s + 2.57·87-s − 1.65·93-s + 0.788·103-s − 1.91·109-s + 0.759·111-s − 1.12·113-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 112896 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5975481403\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5975481403\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 74 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 70 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - 34 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 - 50 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−11.92341925987503222812196334171, −11.17645776453406314067264083133, −10.83315841794481051552790801365, −10.53086373301486406447034963918, −10.03639868347998446712751275956, −9.530936301396733753362194662578, −9.146064291849063580216257462860, −8.556384388953099950017069515726, −8.049301689693412470635436315793, −7.22367243703132774498094031637, −6.77622225013010139485222237452, −6.50659441933642221534056676694, −6.07583110197535462849990361705, −5.40903994981487550417001456711, −4.91959854198531620629975696007, −4.18766244302728721673892190333, −3.65976519563622883803989987662, −2.88037227562263569230672703746, −1.97726333252858067533511947383, −0.57638118451424957264757993507,
0.57638118451424957264757993507, 1.97726333252858067533511947383, 2.88037227562263569230672703746, 3.65976519563622883803989987662, 4.18766244302728721673892190333, 4.91959854198531620629975696007, 5.40903994981487550417001456711, 6.07583110197535462849990361705, 6.50659441933642221534056676694, 6.77622225013010139485222237452, 7.22367243703132774498094031637, 8.049301689693412470635436315793, 8.556384388953099950017069515726, 9.146064291849063580216257462860, 9.530936301396733753362194662578, 10.03639868347998446712751275956, 10.53086373301486406447034963918, 10.83315841794481051552790801365, 11.17645776453406314067264083133, 11.92341925987503222812196334171
