| L(s) = 1 | + 7-s − 7·19-s − 5·25-s + 7·31-s − 37-s − 6·49-s + 12·61-s − 21·67-s + 27·73-s + 21·79-s + 7·103-s − 17·109-s − 11·121-s + 127-s + 131-s − 7·133-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 23·169-s + 173-s − 5·175-s + 179-s + ⋯ |
| L(s) = 1 | + 0.377·7-s − 1.60·19-s − 25-s + 1.25·31-s − 0.164·37-s − 6/7·49-s + 1.53·61-s − 2.56·67-s + 3.16·73-s + 2.36·79-s + 0.689·103-s − 1.62·109-s − 121-s + 0.0887·127-s + 0.0873·131-s − 0.606·133-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.76·169-s + 0.0760·173-s − 0.377·175-s + 0.0747·179-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1016064 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.692415891\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.692415891\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 10 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 - 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
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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
−10.12903857309317230790019540035, −9.800588211176046681211569586413, −9.342799702241428352784788864478, −8.939979610128395540187891016682, −8.434904384505325909384749640394, −8.003864099803725959521632139993, −7.932159118039758247643825473242, −7.25410662625239509677432911629, −6.60416152394509088488982668909, −6.47110121726679888680313449834, −6.00006495241121217889598540346, −5.33992005001489205757925241783, −4.97947098215655556146873351792, −4.41445796866219182384427435668, −4.00442834170983210244022306922, −3.49971805767180016615148989331, −2.74351318089459502351499943417, −2.17743546368480892132760849871, −1.65697134899042216183261283588, −0.59387456950009050273677886087,
0.59387456950009050273677886087, 1.65697134899042216183261283588, 2.17743546368480892132760849871, 2.74351318089459502351499943417, 3.49971805767180016615148989331, 4.00442834170983210244022306922, 4.41445796866219182384427435668, 4.97947098215655556146873351792, 5.33992005001489205757925241783, 6.00006495241121217889598540346, 6.47110121726679888680313449834, 6.60416152394509088488982668909, 7.25410662625239509677432911629, 7.932159118039758247643825473242, 8.003864099803725959521632139993, 8.434904384505325909384749640394, 8.939979610128395540187891016682, 9.342799702241428352784788864478, 9.800588211176046681211569586413, 10.12903857309317230790019540035
