| L(s) = 1 | − 4·4-s − 8·7-s + 10·13-s + 12·16-s + 16·19-s + 32·28-s + 22·31-s − 2·37-s − 26·43-s + 34·49-s − 40·52-s + 28·61-s − 32·64-s + 10·67-s + 34·73-s − 64·76-s − 26·79-s − 80·91-s + 28·97-s − 14·103-s + 34·109-s − 96·112-s − 22·121-s − 88·124-s + 127-s + 131-s − 128·133-s + ⋯ |
| L(s) = 1 | − 2·4-s − 3.02·7-s + 2.77·13-s + 3·16-s + 3.67·19-s + 6.04·28-s + 3.95·31-s − 0.328·37-s − 3.96·43-s + 34/7·49-s − 5.54·52-s + 3.58·61-s − 4·64-s + 1.22·67-s + 3.97·73-s − 7.34·76-s − 2.92·79-s − 8.38·91-s + 2.84·97-s − 1.37·103-s + 3.25·109-s − 9.07·112-s − 2·121-s − 7.90·124-s + 0.0887·127-s + 0.0873·131-s − 11.0·133-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 455625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.067371251\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.067371251\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 2 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )^{2} \) |
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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
−13.18877638682235424909239151398, −12.24502616786121365455911322151, −12.24502616786121365455911322151, −11.40129773178095964858724286650, −11.40129773178095964858724286650, −10.04820957110891225559888247416, −10.04820957110891225559888247416, −9.804478021252617404784341015739, −9.804478021252617404784341015739, −8.831795318502833330339208828788, −8.831795318502833330339208828788, −8.103539044020415994829049592764, −8.103539044020415994829049592764, −6.83497060476461397094513639636, −6.83497060476461397094513639636, −6.01724090331191530541613327241, −6.01724090331191530541613327241, −5.05243608272378186060105532251, −5.05243608272378186060105532251, −3.74647317217248951569851723821, −3.74647317217248951569851723821, −3.13936473638444884836235755024, −3.13936473638444884836235755024, −0.895025331879932514186462794519, −0.895025331879932514186462794519,
0.895025331879932514186462794519, 0.895025331879932514186462794519, 3.13936473638444884836235755024, 3.13936473638444884836235755024, 3.74647317217248951569851723821, 3.74647317217248951569851723821, 5.05243608272378186060105532251, 5.05243608272378186060105532251, 6.01724090331191530541613327241, 6.01724090331191530541613327241, 6.83497060476461397094513639636, 6.83497060476461397094513639636, 8.103539044020415994829049592764, 8.103539044020415994829049592764, 8.831795318502833330339208828788, 8.831795318502833330339208828788, 9.804478021252617404784341015739, 9.804478021252617404784341015739, 10.04820957110891225559888247416, 10.04820957110891225559888247416, 11.40129773178095964858724286650, 11.40129773178095964858724286650, 12.24502616786121365455911322151, 12.24502616786121365455911322151, 13.18877638682235424909239151398
