| L(s) = 1 | − 2·5-s + 2·11-s + 2·13-s + 4·17-s − 6·19-s + 2·25-s + 6·29-s − 16·31-s − 6·37-s − 10·43-s − 16·47-s + 10·49-s − 10·53-s − 4·55-s − 6·59-s + 18·61-s − 4·65-s + 10·67-s − 2·83-s − 8·85-s + 12·95-s − 4·97-s + 22·101-s − 14·107-s − 6·109-s + 12·113-s + 2·121-s + ⋯ |
| L(s) = 1 | − 0.894·5-s + 0.603·11-s + 0.554·13-s + 0.970·17-s − 1.37·19-s + 2/5·25-s + 1.11·29-s − 2.87·31-s − 0.986·37-s − 1.52·43-s − 2.33·47-s + 10/7·49-s − 1.37·53-s − 0.539·55-s − 0.781·59-s + 2.30·61-s − 0.496·65-s + 1.22·67-s − 0.219·83-s − 0.867·85-s + 1.23·95-s − 0.406·97-s + 2.18·101-s − 1.35·107-s − 0.574·109-s + 1.12·113-s + 2/11·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1327104 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.233757388\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.233757388\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 7 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 10 T + 50 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 18 T + 162 T^{2} - 18 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 10 T + 50 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 42 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 2 T + 2 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 162 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−10.00419395783294059633937137675, −9.629298747100390472692533905486, −9.126791310395377687011973647860, −8.533407369549866103537257842390, −8.501326769365921162082958764621, −8.072574652741689627973409719339, −7.52996953182904516445246126827, −7.04834359754263378467797334364, −6.77557927853428421546881480517, −6.28931806461978086200000028298, −5.84800656732092485422756521783, −5.13652814628748907524340912871, −4.98912371406221158588357179407, −4.22013221147723810564784246677, −3.74939197839987385516225497709, −3.51049025110595818772439200611, −2.97519874817941570571558015212, −1.94832630369567704040932942633, −1.59797776481250517815219276734, −0.48934285697923472429924730022,
0.48934285697923472429924730022, 1.59797776481250517815219276734, 1.94832630369567704040932942633, 2.97519874817941570571558015212, 3.51049025110595818772439200611, 3.74939197839987385516225497709, 4.22013221147723810564784246677, 4.98912371406221158588357179407, 5.13652814628748907524340912871, 5.84800656732092485422756521783, 6.28931806461978086200000028298, 6.77557927853428421546881480517, 7.04834359754263378467797334364, 7.52996953182904516445246126827, 8.072574652741689627973409719339, 8.501326769365921162082958764621, 8.533407369549866103537257842390, 9.126791310395377687011973647860, 9.629298747100390472692533905486, 10.00419395783294059633937137675
