| L(s) = 1 | + 2·2-s − 3-s + 2·4-s − 2·6-s + 9-s − 3·11-s − 2·12-s − 2·13-s − 4·16-s + 2·18-s − 6·22-s + 2·23-s + 6·25-s − 4·26-s − 27-s − 8·32-s + 3·33-s + 2·36-s − 4·37-s + 2·39-s − 6·44-s + 4·46-s + 4·47-s + 4·48-s − 10·49-s + 12·50-s − 4·52-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 0.577·3-s + 4-s − 0.816·6-s + 1/3·9-s − 0.904·11-s − 0.577·12-s − 0.554·13-s − 16-s + 0.471·18-s − 1.27·22-s + 0.417·23-s + 6/5·25-s − 0.784·26-s − 0.192·27-s − 1.41·32-s + 0.522·33-s + 1/3·36-s − 0.657·37-s + 0.320·39-s − 0.904·44-s + 0.589·46-s + 0.583·47-s + 0.577·48-s − 1.42·49-s + 1.69·50-s − 0.554·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4752 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.268349092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.268349092\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 3 | $C_1$ | \( 1 + T \) |
| 11 | $C_1$$\times$$C_2$ | \( ( 1 + T )( 1 + 2 T + p T^{2} ) \) |
| good | 5 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 20 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 32 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 53 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 18 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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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
−12.51021995396410622127666731706, −11.87773095456927540890196909399, −11.30969060506094184165204833118, −10.77601207283745221139228907102, −10.17284634700321495333161490449, −9.438573614083758925021986216652, −8.675624784778776503356443039080, −7.86956957485722377619779765629, −6.97714424881146094568630025269, −6.58556032551561385193464285054, −5.57832659259924269412536552346, −5.14824189430156072054815608170, −4.53158781230673150107022365323, −3.48273839851209267437121588412, −2.49772869943614470203581509854,
2.49772869943614470203581509854, 3.48273839851209267437121588412, 4.53158781230673150107022365323, 5.14824189430156072054815608170, 5.57832659259924269412536552346, 6.58556032551561385193464285054, 6.97714424881146094568630025269, 7.86956957485722377619779765629, 8.675624784778776503356443039080, 9.438573614083758925021986216652, 10.17284634700321495333161490449, 10.77601207283745221139228907102, 11.30969060506094184165204833118, 11.87773095456927540890196909399, 12.51021995396410622127666731706
