| L(s) = 1 | + 2·2-s + 3·4-s − 7-s + 4·8-s − 5·9-s − 2·14-s + 5·16-s − 10·18-s + 6·23-s − 10·25-s − 3·28-s + 6·32-s − 15·36-s − 8·37-s − 2·43-s + 12·46-s − 6·49-s − 20·50-s + 12·53-s − 4·56-s + 5·63-s + 7·64-s + 16·67-s − 2·71-s − 20·72-s − 16·74-s + 16·79-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 3/2·4-s − 0.377·7-s + 1.41·8-s − 5/3·9-s − 0.534·14-s + 5/4·16-s − 2.35·18-s + 1.25·23-s − 2·25-s − 0.566·28-s + 1.06·32-s − 5/2·36-s − 1.31·37-s − 0.304·43-s + 1.76·46-s − 6/7·49-s − 2.82·50-s + 1.64·53-s − 0.534·56-s + 0.629·63-s + 7/8·64-s + 1.95·67-s − 0.237·71-s − 2.35·72-s − 1.85·74-s + 1.80·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 988036 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 988036 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 71 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 5 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 41 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 9 T + p T^{2} )( 1 + 9 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 89 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.79510581397524982652968495375, −7.46621865993584119714069747342, −6.68433019836854264455958796314, −6.52013654011009885219318723813, −6.11780053710550114114535207754, −5.38775664071550520628389276206, −5.30767778357765891574349430596, −4.98128679431341487771438855867, −3.95425823104353818397116704374, −3.77653001984849480071749037204, −3.30007748458862767263837345588, −2.51874373869269853485475383946, −2.41081352569639686859829672160, −1.36174454287330615193348852855, 0,
1.36174454287330615193348852855, 2.41081352569639686859829672160, 2.51874373869269853485475383946, 3.30007748458862767263837345588, 3.77653001984849480071749037204, 3.95425823104353818397116704374, 4.98128679431341487771438855867, 5.30767778357765891574349430596, 5.38775664071550520628389276206, 6.11780053710550114114535207754, 6.52013654011009885219318723813, 6.68433019836854264455958796314, 7.46621865993584119714069747342, 7.79510581397524982652968495375
