| L(s) = 1 | + 4-s + 4·7-s − 12·11-s − 3·16-s + 6·17-s + 8·19-s − 12·23-s − 2·25-s + 4·28-s + 12·29-s + 4·31-s − 12·44-s − 2·49-s + 18·53-s − 7·64-s − 28·67-s + 6·68-s + 8·76-s − 48·77-s − 12·83-s + 30·89-s − 12·92-s + 22·97-s − 2·100-s + 20·103-s + 10·109-s − 12·112-s + ⋯ |
| L(s) = 1 | + 1/2·4-s + 1.51·7-s − 3.61·11-s − 3/4·16-s + 1.45·17-s + 1.83·19-s − 2.50·23-s − 2/5·25-s + 0.755·28-s + 2.22·29-s + 0.718·31-s − 1.80·44-s − 2/7·49-s + 2.47·53-s − 7/8·64-s − 3.42·67-s + 0.727·68-s + 0.917·76-s − 5.47·77-s − 1.31·83-s + 3.17·89-s − 1.25·92-s + 2.23·97-s − 1/5·100-s + 1.97·103-s + 0.957·109-s − 1.13·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700569 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.004253008\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.004253008\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 31 | $C_2$ | \( 1 - 4 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - T^{2} + p^{2} T^{4} \) |
| 5 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 59 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 67 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 - 115 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 139 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 38 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 + 85 T^{2} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 15 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
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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
−10.47184164577259266548818709096, −10.10012789257600988694109235924, −10.04095761900829772504785074318, −9.052745581040799837705992198639, −8.520839810989434246157083558131, −8.106166437703895138642426506825, −7.74255942600407098987049408045, −7.61054370239988270680216461347, −7.42955838027674491043567141789, −6.43149963698056450655360523081, −5.84058952130104413387825988706, −5.60339733845433149654577089609, −5.00316458534270154566652206252, −4.83794736382517457983396084500, −4.28857205528107958071506411829, −3.18998912768154095196843225304, −2.93011503847858510737829660415, −2.28477903326300886773087759210, −1.81274990696982997293394415470, −0.67931400258620233856188265920,
0.67931400258620233856188265920, 1.81274990696982997293394415470, 2.28477903326300886773087759210, 2.93011503847858510737829660415, 3.18998912768154095196843225304, 4.28857205528107958071506411829, 4.83794736382517457983396084500, 5.00316458534270154566652206252, 5.60339733845433149654577089609, 5.84058952130104413387825988706, 6.43149963698056450655360523081, 7.42955838027674491043567141789, 7.61054370239988270680216461347, 7.74255942600407098987049408045, 8.106166437703895138642426506825, 8.520839810989434246157083558131, 9.052745581040799837705992198639, 10.04095761900829772504785074318, 10.10012789257600988694109235924, 10.47184164577259266548818709096
