| L(s) = 1 | − 3·13-s − 8·19-s + 5·25-s − 22·31-s + 21·43-s − 13·49-s − 13·61-s + 5·67-s − 7·73-s + 13·79-s − 24·97-s − 26·103-s − 36·109-s + 22·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s − 7·169-s + 173-s + 179-s + 181-s + ⋯ |
| L(s) = 1 | − 0.832·13-s − 1.83·19-s + 25-s − 3.95·31-s + 3.20·43-s − 1.85·49-s − 1.66·61-s + 0.610·67-s − 0.819·73-s + 1.46·79-s − 2.43·97-s − 2.56·103-s − 3.44·109-s + 2·121-s + 0.0887·127-s + 0.0873·131-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 − 0.538·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7485696 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5071099064\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5071099064\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 8 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 11 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 11 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - 8 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 - T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 11 T + p T^{2} ) \) |
| 71 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 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 | $C_2$ | \( ( 1 + 5 T + p T^{2} )( 1 + 19 T + p T^{2} ) \) |
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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
−9.217018669627786751824778410137, −8.608041226420124875172981369640, −8.333455060210173564403718504570, −7.75217281028595077847371960168, −7.58818215180578250457858881005, −7.00191865631041658241799674209, −6.91612203668815252784796650502, −6.36747841830732286112121533392, −5.86049083056930808109416911038, −5.57963501553589169832975105342, −5.19376121310791619633665344114, −4.59671503821827805616811907869, −4.35435377704753008912613443546, −3.80086640690622082520208224821, −3.50233521887085029830290537441, −2.69222629766550002500080713043, −2.47416338839528205052242930634, −1.82453035333257578499902791404, −1.36326145107949271972607416332, −0.22428242819769975415970190403,
0.22428242819769975415970190403, 1.36326145107949271972607416332, 1.82453035333257578499902791404, 2.47416338839528205052242930634, 2.69222629766550002500080713043, 3.50233521887085029830290537441, 3.80086640690622082520208224821, 4.35435377704753008912613443546, 4.59671503821827805616811907869, 5.19376121310791619633665344114, 5.57963501553589169832975105342, 5.86049083056930808109416911038, 6.36747841830732286112121533392, 6.91612203668815252784796650502, 7.00191865631041658241799674209, 7.58818215180578250457858881005, 7.75217281028595077847371960168, 8.333455060210173564403718504570, 8.608041226420124875172981369640, 9.217018669627786751824778410137
