| L(s) = 1 | + 2·3-s − 4-s + 2·5-s + 9-s − 2·12-s + 4·15-s − 3·16-s − 2·20-s + 3·25-s − 4·27-s + 14·31-s − 36-s − 16·37-s + 2·45-s + 18·47-s − 6·48-s + 4·49-s + 6·53-s − 12·59-s − 4·60-s + 7·64-s + 2·67-s + 6·71-s + 6·75-s − 6·80-s − 11·81-s − 6·89-s + ⋯ |
| L(s) = 1 | + 1.15·3-s − 1/2·4-s + 0.894·5-s + 1/3·9-s − 0.577·12-s + 1.03·15-s − 3/4·16-s − 0.447·20-s + 3/5·25-s − 0.769·27-s + 2.51·31-s − 1/6·36-s − 2.63·37-s + 0.298·45-s + 2.62·47-s − 0.866·48-s + 4/7·49-s + 0.824·53-s − 1.56·59-s − 0.516·60-s + 7/8·64-s + 0.244·67-s + 0.712·71-s + 0.692·75-s − 0.670·80-s − 1.22·81-s − 0.635·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3294225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.383274952\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.383274952\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| 5 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 14 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 28 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 16 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$$\times$$C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 - 6 T + p T^{2} ) \) |
| 53 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 59 | $C_2$$\times$$C_2$ | \( ( 1 + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 61 | $C_2^2$ | \( 1 - 106 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2$$\times$$C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 71 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 73 | $C_2^2$ | \( 1 + 122 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 - 62 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 10 T + p T^{2} ) \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.49098539101166052381138121357, −7.14703542001175774353519888820, −6.74818164566120742574663576809, −6.29509018904608682994380414516, −5.83306727345807364409216570044, −5.39118208869250254857641721193, −4.97205368792216796302593140136, −4.47309452929910871749352060787, −4.03150232891126431605696030941, −3.57372290050761653531974714460, −2.92612840452621368318519794069, −2.57507624467624512088497102936, −2.12891273114152411316998270488, −1.51135865206012438638246386158, −0.63628903793762127103374436332,
0.63628903793762127103374436332, 1.51135865206012438638246386158, 2.12891273114152411316998270488, 2.57507624467624512088497102936, 2.92612840452621368318519794069, 3.57372290050761653531974714460, 4.03150232891126431605696030941, 4.47309452929910871749352060787, 4.97205368792216796302593140136, 5.39118208869250254857641721193, 5.83306727345807364409216570044, 6.29509018904608682994380414516, 6.74818164566120742574663576809, 7.14703542001175774353519888820, 7.49098539101166052381138121357
