| L(s) = 1 | − 16·5-s + 14·7-s − 4·11-s + 52·13-s − 56·17-s + 32·19-s + 44·23-s + 114·25-s − 20·29-s − 296·31-s − 224·35-s + 172·37-s − 224·41-s − 144·43-s + 344·47-s + 147·49-s − 364·53-s + 64·55-s − 464·59-s + 620·61-s − 832·65-s + 904·67-s − 508·71-s + 12·73-s − 56·77-s + 344·79-s − 280·83-s + ⋯ |
| L(s) = 1 | − 1.43·5-s + 0.755·7-s − 0.109·11-s + 1.10·13-s − 0.798·17-s + 0.386·19-s + 0.398·23-s + 0.911·25-s − 0.128·29-s − 1.71·31-s − 1.08·35-s + 0.764·37-s − 0.853·41-s − 0.510·43-s + 1.06·47-s + 3/7·49-s − 0.943·53-s + 0.156·55-s − 1.02·59-s + 1.30·61-s − 1.58·65-s + 1.64·67-s − 0.849·71-s + 0.0192·73-s − 0.0828·77-s + 0.489·79-s − 0.370·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4064256 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.431966618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.431966618\) |
| \(L(\frac{5}{2})\) |
|
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 | | \( 1 \) |
| 3 | | \( 1 \) |
| 7 | $C_1$ | \( ( 1 - p T )^{2} \) |
| good | 5 | $D_{4}$ | \( 1 + 16 T + 142 T^{2} + 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 2494 T^{2} + 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 p T + p^{3} T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 56 T + 9062 T^{2} + 56 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 - 32 T + 13286 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 44 T + 4006 T^{2} - 44 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 20 T + 48190 T^{2} + 20 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 296 T + 64286 T^{2} + 296 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 172 T + 64670 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 224 T + 129574 T^{2} + 224 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 144 T + 120166 T^{2} + 144 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 344 T + 231038 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 364 T + 286846 T^{2} + 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 464 T + 265750 T^{2} + 464 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 620 T + 494334 T^{2} - 620 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 904 T + 651030 T^{2} - 904 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 508 T + 544870 T^{2} + 508 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 12 T + 276518 T^{2} - 12 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 344 T + 1009470 T^{2} - 344 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 280 T + 458662 T^{2} + 280 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 848 T + 1585414 T^{2} + 848 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 1372 T + 2047574 T^{2} - 1372 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.785483750257138644797016148852, −8.615213719745439528597324452137, −8.050713684796760933055290976606, −8.032992299398302150001205219802, −7.33486261393006034677382199726, −7.26683141500132443280490917019, −6.66288804059122264444370555568, −6.37733861564645880213825527137, −5.57955756307613318168904279524, −5.51370126356792532569755908819, −4.86251021801240574766963152994, −4.47825830911016504842179357620, −3.92771345377283835899129214304, −3.84242656745767582426718352214, −3.16073284888187926908054801859, −2.82297340268753138202308723683, −1.82650965861004912347207805913, −1.73309803815148815677002497422, −0.76069146623711795644219249290, −0.44247878753703232318866794002,
0.44247878753703232318866794002, 0.76069146623711795644219249290, 1.73309803815148815677002497422, 1.82650965861004912347207805913, 2.82297340268753138202308723683, 3.16073284888187926908054801859, 3.84242656745767582426718352214, 3.92771345377283835899129214304, 4.47825830911016504842179357620, 4.86251021801240574766963152994, 5.51370126356792532569755908819, 5.57955756307613318168904279524, 6.37733861564645880213825527137, 6.66288804059122264444370555568, 7.26683141500132443280490917019, 7.33486261393006034677382199726, 8.032992299398302150001205219802, 8.050713684796760933055290976606, 8.615213719745439528597324452137, 8.785483750257138644797016148852
