| L(s) = 1 | − 4·4-s − 72·11-s + 16·16-s + 182·19-s + 552·29-s + 382·31-s − 120·41-s + 288·44-s + 565·49-s + 1.48e3·59-s + 334·61-s − 64·64-s − 1.17e3·71-s − 728·76-s − 328·79-s + 2.49e3·89-s − 888·101-s + 1.93e3·109-s − 2.20e3·116-s + 1.22e3·121-s − 1.52e3·124-s + ⋯ |
| L(s) = 1 | − 1/2·4-s − 1.97·11-s + 1/4·16-s + 2.19·19-s + 3.53·29-s + 2.21·31-s − 0.457·41-s + 0.986·44-s + 1.64·49-s + 3.28·59-s + 0.701·61-s − 1/8·64-s − 1.96·71-s − 1.09·76-s − 0.467·79-s + 2.97·89-s − 0.874·101-s + 1.69·109-s − 1.76·116-s + 0.921·121-s − 1.10·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 202500 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.713562116\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.713562116\) |
| \(L(\frac{5}{2})\) |
|
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 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| 3 | | \( 1 \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2^2$ | \( 1 - 565 T^{2} + p^{6} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + 36 T + p^{3} T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 4105 T^{2} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 9682 T^{2} + p^{6} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 91 T + p^{3} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 20734 T^{2} + p^{6} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 276 T + p^{3} T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 191 T + p^{3} T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 36790 T^{2} + p^{6} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 60 T + p^{3} T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 156613 T^{2} + p^{6} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 152354 T^{2} + p^{6} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 76790 T^{2} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 744 T + p^{3} T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 167 T + p^{3} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 392677 T^{2} + p^{6} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 588 T + p^{3} T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 162866 T^{2} + p^{6} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 164 T + p^{3} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 659158 T^{2} + p^{6} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 1248 T + p^{3} T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 - 617545 T^{2} + p^{6} T^{4} \) |
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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
−10.62284928853726417795088437891, −10.29841567752806645979354562846, −9.987175152020956575362982121629, −9.934978587026581263393392126340, −8.911276888816445216283732774295, −8.664658720501297887862065302921, −8.091122402164440803502143178906, −7.88638865130561664464652003333, −7.26469945981216460894488234391, −6.78293055421717022132271180181, −6.18710203317889426590344945192, −5.49338182863879290846114428314, −5.16380232539005393820449772736, −4.74700347141400335362808966950, −4.19482502408834331661313655628, −3.22762415154801947909160522475, −2.83164762257168451306261265075, −2.38164170879179700850593233672, −0.929758947206530202672196434107, −0.73564221042227265882887672263,
0.73564221042227265882887672263, 0.929758947206530202672196434107, 2.38164170879179700850593233672, 2.83164762257168451306261265075, 3.22762415154801947909160522475, 4.19482502408834331661313655628, 4.74700347141400335362808966950, 5.16380232539005393820449772736, 5.49338182863879290846114428314, 6.18710203317889426590344945192, 6.78293055421717022132271180181, 7.26469945981216460894488234391, 7.88638865130561664464652003333, 8.091122402164440803502143178906, 8.664658720501297887862065302921, 8.911276888816445216283732774295, 9.934978587026581263393392126340, 9.987175152020956575362982121629, 10.29841567752806645979354562846, 10.62284928853726417795088437891
