| L(s) = 1 | + 4·11-s − 10·19-s + 20·29-s + 6·31-s + 16·41-s + 5·49-s + 20·59-s + 14·61-s − 16·71-s − 24·101-s − 10·109-s − 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 25·169-s + 173-s + 179-s + 181-s + 191-s + 193-s + ⋯ |
| L(s) = 1 | + 1.20·11-s − 2.29·19-s + 3.71·29-s + 1.07·31-s + 2.49·41-s + 5/7·49-s + 2.60·59-s + 1.79·61-s − 1.89·71-s − 2.38·101-s − 0.957·109-s − 0.909·121-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 + 1.92·169-s + 0.0760·173-s + 0.0747·179-s + 0.0743·181-s + 0.0723·191-s + 0.0719·193-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12960000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.634089348\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.634089348\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 5 T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 25 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 90 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 125 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 130 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 95 T^{2} + p^{2} T^{4} \) |
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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
−8.510207643854788456513406531326, −8.457189917590419148981404775732, −8.190273255472999823568622119857, −7.75728981447324310512869143578, −6.93293316396133459635276734796, −6.91688036260201372110304281150, −6.44051054470360246201953084979, −6.41229401047564575705178209281, −5.67703057570696635477832237119, −5.54197284861683970724801721398, −4.76930070963763838901387465898, −4.34905375740304426911335785130, −4.09325649730811148522801939385, −4.09179695764320897983563816891, −3.01440658992017247458849886765, −2.80907327913162999350199238449, −2.32283747100326917277925455106, −1.80180902406788749030214493980, −0.883639595464901708145119393493, −0.77438724675417013692147743685,
0.77438724675417013692147743685, 0.883639595464901708145119393493, 1.80180902406788749030214493980, 2.32283747100326917277925455106, 2.80907327913162999350199238449, 3.01440658992017247458849886765, 4.09179695764320897983563816891, 4.09325649730811148522801939385, 4.34905375740304426911335785130, 4.76930070963763838901387465898, 5.54197284861683970724801721398, 5.67703057570696635477832237119, 6.41229401047564575705178209281, 6.44051054470360246201953084979, 6.91688036260201372110304281150, 6.93293316396133459635276734796, 7.75728981447324310512869143578, 8.190273255472999823568622119857, 8.457189917590419148981404775732, 8.510207643854788456513406531326
