| L(s) = 1 | − 5·3-s + 16·9-s + 20·13-s − 14·19-s − 35·27-s − 84·31-s − 80·37-s − 100·39-s + 100·43-s − 98·49-s + 70·57-s − 16·61-s − 90·67-s − 70·73-s − 24·79-s + 31·81-s + 420·93-s − 140·97-s + 140·103-s − 176·109-s + 400·111-s + 320·117-s − 33·121-s + 127-s − 500·129-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 5/3·3-s + 16/9·9-s + 1.53·13-s − 0.736·19-s − 1.29·27-s − 2.70·31-s − 2.16·37-s − 2.56·39-s + 2.32·43-s − 2·49-s + 1.22·57-s − 0.262·61-s − 1.34·67-s − 0.958·73-s − 0.303·79-s + 0.382·81-s + 4.51·93-s − 1.44·97-s + 1.35·103-s − 1.61·109-s + 3.60·111-s + 2.73·117-s − 0.272·121-s + 0.00787·127-s − 3.87·129-s + 0.00763·131-s + 0.00729·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440000 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.09912610536\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.09912610536\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 + 5 T + p^{2} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 + 3 p T^{2} + p^{4} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 10 T + p^{2} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 567 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 7 T + p^{2} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 662 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 582 T^{2} + p^{4} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 42 T + p^{2} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 40 T + p^{2} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 3087 T^{2} + p^{4} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 50 T + p^{2} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2262 T^{2} + p^{4} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 3462 T^{2} + p^{4} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 2562 T^{2} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 45 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 8982 T^{2} + p^{4} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 35 T + p^{2} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 12 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 8927 T^{2} + p^{4} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 97 T + p^{2} T^{2} )( 1 + 97 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 + 70 T + p^{2} T^{2} )^{2} \) |
| 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
−10.06080987981484462435677213920, −9.104840705362244004613420630956, −9.104046073387425838131259919499, −8.709639531485777815012602388886, −8.070271355550119594556357116038, −7.49069277738582295931583928480, −7.31657121709517981562226636831, −6.56371296774141269459060328706, −6.51660632905670742941123477004, −5.80617628543334828309934118988, −5.75075852927729883882626936139, −5.14866743377138493509559222844, −4.79452945446858682497260788802, −4.02521801472217760338777038469, −3.86249104009996501345695392145, −3.26425557286512377540173694717, −2.37602536304731775799188247711, −1.42142824367537163018304147781, −1.40654623691832487618054811898, −0.10996348351007812901318932232,
0.10996348351007812901318932232, 1.40654623691832487618054811898, 1.42142824367537163018304147781, 2.37602536304731775799188247711, 3.26425557286512377540173694717, 3.86249104009996501345695392145, 4.02521801472217760338777038469, 4.79452945446858682497260788802, 5.14866743377138493509559222844, 5.75075852927729883882626936139, 5.80617628543334828309934118988, 6.51660632905670742941123477004, 6.56371296774141269459060328706, 7.31657121709517981562226636831, 7.49069277738582295931583928480, 8.070271355550119594556357116038, 8.709639531485777815012602388886, 9.104046073387425838131259919499, 9.104840705362244004613420630956, 10.06080987981484462435677213920
