| L(s) = 1 | + 3·4-s − 4·5-s + 5·16-s − 12·20-s + 11·25-s − 16·29-s + 8·31-s − 8·41-s + 16·59-s − 20·61-s + 3·64-s − 32·71-s + 16·79-s − 20·80-s − 24·89-s + 33·100-s + 24·101-s − 4·109-s − 48·116-s − 22·121-s + 24·124-s − 24·125-s + 127-s + 131-s + 137-s + 139-s + 64·145-s + ⋯ |
| L(s) = 1 | + 3/2·4-s − 1.78·5-s + 5/4·16-s − 2.68·20-s + 11/5·25-s − 2.97·29-s + 1.43·31-s − 1.24·41-s + 2.08·59-s − 2.56·61-s + 3/8·64-s − 3.79·71-s + 1.80·79-s − 2.23·80-s − 2.54·89-s + 3.29·100-s + 2.38·101-s − 0.383·109-s − 4.45·116-s − 2·121-s + 2.15·124-s − 2.14·125-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 5.31·145-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.354562637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.354562637\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - 3 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 2 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 90 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 178 T^{2} + p^{2} T^{4} \) |
| 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
−9.092868793687922157592853116534, −8.718843900624781611652744187688, −8.479308093113154377806674975973, −7.77650686555723543482984214618, −7.70603207539630724440698662490, −7.36108990337956316994318630997, −7.10122067147964818603040738007, −6.45564615733284198659467677511, −6.42453843286191448401677044432, −5.62902707213128541853992599041, −5.45990877592090270323746026648, −4.69436286195869808609019078381, −4.37986964796888691500592779235, −3.79907987556961211025408842975, −3.51822261343081351139452541841, −2.93383144340744471385992259118, −2.66671922835798334545929509134, −1.80870983257075546604533392253, −1.44952737206219549080872734068, −0.39093187073617258248755246754,
0.39093187073617258248755246754, 1.44952737206219549080872734068, 1.80870983257075546604533392253, 2.66671922835798334545929509134, 2.93383144340744471385992259118, 3.51822261343081351139452541841, 3.79907987556961211025408842975, 4.37986964796888691500592779235, 4.69436286195869808609019078381, 5.45990877592090270323746026648, 5.62902707213128541853992599041, 6.42453843286191448401677044432, 6.45564615733284198659467677511, 7.10122067147964818603040738007, 7.36108990337956316994318630997, 7.70603207539630724440698662490, 7.77650686555723543482984214618, 8.479308093113154377806674975973, 8.718843900624781611652744187688, 9.092868793687922157592853116534
