| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s + 2·12-s + 8·13-s + 14-s + 16-s + 12·17-s + 18-s − 4·19-s + 2·21-s + 2·24-s − 10·25-s + 8·26-s − 4·27-s + 28-s + 12·29-s + 32-s + 12·34-s + 36-s − 4·38-s + 16·39-s + 12·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s + 2.21·13-s + 0.267·14-s + 1/4·16-s + 2.91·17-s + 0.235·18-s − 0.917·19-s + 0.436·21-s + 0.408·24-s − 2·25-s + 1.56·26-s − 0.769·27-s + 0.188·28-s + 2.22·29-s + 0.176·32-s + 2.05·34-s + 1/6·36-s − 0.648·38-s + 2.56·39-s + 1.87·41-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
| \(L(1)\) |
\(\approx\) |
\(5.451760094\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.451760094\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( 1 - T \) |
| 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 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 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
−8.245010328271296459357780366393, −8.222985047096640302551257734717, −7.87021772021814993834479658639, −7.64010214491199898083268392854, −6.47781377385923123584313505983, −6.37349457460642388728354325328, −5.88407913223931048395780461484, −5.36663802373794330843690451326, −4.74683922157615295493069282209, −3.93522119575008455702519349301, −3.79391663707115759548696099652, −3.07302685938302839530976846531, −2.83833445427180997893704571146, −1.64950502134773404742708641242, −1.33827299261372059142982522511,
1.33827299261372059142982522511, 1.64950502134773404742708641242, 2.83833445427180997893704571146, 3.07302685938302839530976846531, 3.79391663707115759548696099652, 3.93522119575008455702519349301, 4.74683922157615295493069282209, 5.36663802373794330843690451326, 5.88407913223931048395780461484, 6.37349457460642388728354325328, 6.47781377385923123584313505983, 7.64010214491199898083268392854, 7.87021772021814993834479658639, 8.222985047096640302551257734717, 8.245010328271296459357780366393
