| L(s) = 1 | − 2·3-s + 4-s + 7-s + 9-s − 2·12-s + 16-s + 12·17-s − 2·21-s − 10·25-s + 4·27-s + 28-s + 36-s + 4·37-s + 12·41-s + 16·43-s − 24·47-s − 2·48-s + 49-s − 24·51-s − 12·59-s + 63-s + 64-s − 8·67-s + 12·68-s + 20·75-s + 16·79-s − 11·81-s + ⋯ |
| L(s) = 1 | − 1.15·3-s + 1/2·4-s + 0.377·7-s + 1/3·9-s − 0.577·12-s + 1/4·16-s + 2.91·17-s − 0.436·21-s − 2·25-s + 0.769·27-s + 0.188·28-s + 1/6·36-s + 0.657·37-s + 1.87·41-s + 2.43·43-s − 3.50·47-s − 0.288·48-s + 1/7·49-s − 3.36·51-s − 1.56·59-s + 0.125·63-s + 1/8·64-s − 0.977·67-s + 1.45·68-s + 2.30·75-s + 1.80·79-s − 1.22·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12348 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12348 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8566614515\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8566614515\) |
| \(L(\frac{3}{2})\) |
|
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_1$$\times$$C_1$ | \( ( 1 - T )( 1 + T ) \) |
| 3 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | $C_1$ | \( 1 - T \) |
| good | 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 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
−11.23136141438460806904855801814, −11.03381616069283788346556339454, −10.27339907123230780461643050534, −9.765547119459919407856461234632, −9.393998538973797562285424645673, −8.245010328271296459357780366393, −7.64010214491199898083268392854, −7.57571100088867902110310233811, −6.37349457460642388728354325328, −5.88407913223931048395780461484, −5.57928681742950427486583645839, −4.74683922157615295493069282209, −3.79391663707115759548696099652, −2.83833445427180997893704571146, −1.33827299261372059142982522511,
1.33827299261372059142982522511, 2.83833445427180997893704571146, 3.79391663707115759548696099652, 4.74683922157615295493069282209, 5.57928681742950427486583645839, 5.88407913223931048395780461484, 6.37349457460642388728354325328, 7.57571100088867902110310233811, 7.64010214491199898083268392854, 8.245010328271296459357780366393, 9.393998538973797562285424645673, 9.765547119459919407856461234632, 10.27339907123230780461643050534, 11.03381616069283788346556339454, 11.23136141438460806904855801814
