| L(s) = 1 | − 4·3-s − 4·5-s + 6·9-s + 2·11-s + 16·15-s + 8·23-s + 2·25-s + 4·27-s − 8·33-s − 20·37-s − 24·45-s − 16·47-s + 2·49-s + 12·53-s − 8·55-s − 28·59-s − 20·67-s − 32·69-s + 24·71-s − 8·75-s − 37·81-s − 4·89-s − 4·97-s + 12·99-s − 8·103-s + 80·111-s + 4·113-s + ⋯ |
| L(s) = 1 | − 2.30·3-s − 1.78·5-s + 2·9-s + 0.603·11-s + 4.13·15-s + 1.66·23-s + 2/5·25-s + 0.769·27-s − 1.39·33-s − 3.28·37-s − 3.57·45-s − 2.33·47-s + 2/7·49-s + 1.64·53-s − 1.07·55-s − 3.64·59-s − 2.44·67-s − 3.85·69-s + 2.84·71-s − 0.923·75-s − 4.11·81-s − 0.423·89-s − 0.406·97-s + 1.20·99-s − 0.788·103-s + 7.59·111-s + 0.376·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1982464 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 11 | $C_2$ | \( 1 - 2 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 5 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 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
−7.11986921553082558730095252140, −6.97464122003923989664965494039, −6.33050260025483473718379487161, −6.21482808880331535088661500986, −5.51628661112943844767821861262, −5.14111388305651036139020256500, −4.84373431553687523132526670248, −4.44779545148279820066365092430, −3.80605036007969937328079924764, −3.36730269661969347724618284285, −2.95202961519500398922987679691, −1.71589136076445568525690834817, −1.05257364243843126849176102869, 0, 0,
1.05257364243843126849176102869, 1.71589136076445568525690834817, 2.95202961519500398922987679691, 3.36730269661969347724618284285, 3.80605036007969937328079924764, 4.44779545148279820066365092430, 4.84373431553687523132526670248, 5.14111388305651036139020256500, 5.51628661112943844767821861262, 6.21482808880331535088661500986, 6.33050260025483473718379487161, 6.97464122003923989664965494039, 7.11986921553082558730095252140
