| L(s) = 1 | − 6·11-s − 8·13-s − 2·25-s − 4·37-s + 24·47-s + 2·49-s − 30·59-s − 16·61-s − 12·71-s − 22·73-s + 24·83-s + 26·97-s − 6·107-s − 8·109-s + 5·121-s + 127-s + 131-s + 137-s + 139-s + 48·143-s + 149-s + 151-s + 157-s + 163-s + 167-s + 22·169-s + 173-s + ⋯ |
| L(s) = 1 | − 1.80·11-s − 2.21·13-s − 2/5·25-s − 0.657·37-s + 3.50·47-s + 2/7·49-s − 3.90·59-s − 2.04·61-s − 1.42·71-s − 2.57·73-s + 2.63·83-s + 2.63·97-s − 0.580·107-s − 0.766·109-s + 5/11·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 4.01·143-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 1.69·169-s + 0.0760·173-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 26873856 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 31 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 35 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 - 146 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 14 T^{2} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 13 T + p T^{2} )^{2} \) |
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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
−7.67350903779401724855999508825, −7.62611704322947138936574020204, −7.50441035974455279455752874791, −7.27072892108541548380484851568, −6.57704499976311611594225050833, −6.12541428046831450641656521625, −5.89844809057486288426593256482, −5.29541209981851289314870824777, −5.25729890067862831688816183483, −4.69527357879641793342988873957, −4.38116493385442532726123958805, −4.17793356059083870042674719010, −3.16108781818622316317540530935, −3.07815950505512556271912172336, −2.69308387970470332083767621438, −2.08465067806497984259645401232, −1.92662980715842389527602360838, −0.985005037981796487380401561060, 0, 0,
0.985005037981796487380401561060, 1.92662980715842389527602360838, 2.08465067806497984259645401232, 2.69308387970470332083767621438, 3.07815950505512556271912172336, 3.16108781818622316317540530935, 4.17793356059083870042674719010, 4.38116493385442532726123958805, 4.69527357879641793342988873957, 5.25729890067862831688816183483, 5.29541209981851289314870824777, 5.89844809057486288426593256482, 6.12541428046831450641656521625, 6.57704499976311611594225050833, 7.27072892108541548380484851568, 7.50441035974455279455752874791, 7.62611704322947138936574020204, 7.67350903779401724855999508825
