| L(s) = 1 | + 4·4-s + 2·11-s + 12·16-s + 14·19-s − 12·29-s − 14·31-s + 12·41-s + 8·44-s + 13·49-s + 10·61-s + 32·64-s + 24·71-s + 56·76-s + 8·79-s + 12·89-s − 22·109-s − 48·116-s + 3·121-s − 56·124-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + ⋯ |
| L(s) = 1 | + 2·4-s + 0.603·11-s + 3·16-s + 3.21·19-s − 2.22·29-s − 2.51·31-s + 1.87·41-s + 1.20·44-s + 13/7·49-s + 1.28·61-s + 4·64-s + 2.84·71-s + 6.42·76-s + 0.900·79-s + 1.27·89-s − 2.10·109-s − 4.45·116-s + 3/11·121-s − 5.02·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6125625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(6.074195548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.074195548\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 2 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 7 | $C_2^2$ | \( 1 - 13 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 109 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−9.106151442260302025402979844486, −9.034780048850486020261776817385, −7.931761786785549635006573289056, −7.87355519110775295375839220454, −7.48448661693733516914137586594, −7.32945710960031162431914299841, −6.76532272327669584285943252451, −6.69590727417759405042490080706, −5.87865026222168586660695894397, −5.67037405981371547104126721338, −5.30096916079741406154216323972, −5.21455883905553777613422073597, −3.88467657385001751869006516793, −3.85669466932834557595154315641, −3.40712924817309745395910884968, −2.93729826510640535921209487457, −2.21591638614725870500589481145, −2.07194957795591154620869899081, −1.25845262176396434545503229591, −0.894032717299712361175198817605,
0.894032717299712361175198817605, 1.25845262176396434545503229591, 2.07194957795591154620869899081, 2.21591638614725870500589481145, 2.93729826510640535921209487457, 3.40712924817309745395910884968, 3.85669466932834557595154315641, 3.88467657385001751869006516793, 5.21455883905553777613422073597, 5.30096916079741406154216323972, 5.67037405981371547104126721338, 5.87865026222168586660695894397, 6.69590727417759405042490080706, 6.76532272327669584285943252451, 7.32945710960031162431914299841, 7.48448661693733516914137586594, 7.87355519110775295375839220454, 7.931761786785549635006573289056, 9.034780048850486020261776817385, 9.106151442260302025402979844486
