L(s) = 1 | + 4-s − 2·9-s − 2·11-s + 2·29-s − 2·36-s − 2·44-s − 49-s − 64-s − 2·71-s + 2·79-s + 3·81-s + 4·99-s + 2·109-s + 2·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + ⋯ |
L(s) = 1 | + 4-s − 2·9-s − 2·11-s + 2·29-s − 2·36-s − 2·44-s − 49-s − 64-s − 2·71-s + 2·79-s + 3·81-s + 4·99-s + 2·109-s + 2·116-s + 121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 2·169-s + 173-s + 179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 30625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4646724337\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4646724337\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
---|
bad | 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
good | 2 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 3 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 19 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 23 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 37 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 47 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 61 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 67 | $C_2^2$ | \( 1 - T^{2} + T^{4} \) |
| 71 | $C_2$ | \( ( 1 + T + T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - T + T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - T )^{2}( 1 + T )^{2} \) |
| 97 | $C_2$ | \( ( 1 + T^{2} )^{2} \) |
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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
−13.51079718904401974351850817746, −12.53910350627727582540120298789, −12.14444489939289199289128893931, −11.70372483322166441461992033855, −11.03055908768080276870833170769, −11.01720550911740402855652502622, −10.29555842557836072172498395314, −9.967080819986157002800541397073, −8.948939647607505608978697134519, −8.660735445337753490462854231169, −7.892558870453853779324276599677, −7.82270203266143885773618920167, −6.90392282848403870644246039249, −6.28445743743243721438375152643, −5.86088782635053946864062341182, −5.16453359959286448991575163680, −4.70197917174654042535704225276, −3.27454930028603520033152893804, −2.78551367007755798973085985436, −2.28332171512070147804322165723,
2.28332171512070147804322165723, 2.78551367007755798973085985436, 3.27454930028603520033152893804, 4.70197917174654042535704225276, 5.16453359959286448991575163680, 5.86088782635053946864062341182, 6.28445743743243721438375152643, 6.90392282848403870644246039249, 7.82270203266143885773618920167, 7.892558870453853779324276599677, 8.660735445337753490462854231169, 8.948939647607505608978697134519, 9.967080819986157002800541397073, 10.29555842557836072172498395314, 11.01720550911740402855652502622, 11.03055908768080276870833170769, 11.70372483322166441461992033855, 12.14444489939289199289128893931, 12.53910350627727582540120298789, 13.51079718904401974351850817746