| L(s) = 1 | − 2-s − 11·4-s + 10·5-s + 15·8-s − 10·10-s + 4·11-s + 22·13-s + 61·16-s − 58·17-s − 110·20-s − 4·22-s + 82·23-s + 75·25-s − 22·26-s + 334·29-s + 210·31-s − 89·32-s + 58·34-s + 6·37-s + 150·40-s − 176·41-s + 46·43-s − 44·44-s − 82·46-s − 514·47-s − 75·50-s − 242·52-s + ⋯ |
| L(s) = 1 | − 0.353·2-s − 1.37·4-s + 0.894·5-s + 0.662·8-s − 0.316·10-s + 0.109·11-s + 0.469·13-s + 0.953·16-s − 0.827·17-s − 1.22·20-s − 0.0387·22-s + 0.743·23-s + 3/5·25-s − 0.165·26-s + 2.13·29-s + 1.21·31-s − 0.491·32-s + 0.292·34-s + 0.0266·37-s + 0.592·40-s − 0.670·41-s + 0.163·43-s − 0.150·44-s − 0.262·46-s − 1.59·47-s − 0.212·50-s − 0.645·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4862025 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.840180887\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.840180887\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | | \( 1 \) |
| 5 | $C_1$ | \( ( 1 - p T )^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $D_{4}$ | \( 1 + T + 3 p^{2} T^{2} + p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 4 T + 2598 T^{2} - 4 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 22 T + 2458 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 58 T + 7794 T^{2} + 58 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 5490 T^{2} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 82 T + 25182 T^{2} - 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 334 T + 72842 T^{2} - 334 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 210 T + 69230 T^{2} - 210 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 - 6 T + 97490 T^{2} - 6 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 176 T + 134094 T^{2} + 176 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 46 T + 42430 T^{2} - 46 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 514 T + 269870 T^{2} + 514 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 808 T + 449478 T^{2} - 808 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + 284 T + p^{3} T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 618 T + 515018 T^{2} - 618 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 694 T + 681118 T^{2} - 694 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 814 T + 769934 T^{2} - 814 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 82 T + 422290 T^{2} + 82 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 600 T + 618846 T^{2} - 600 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 268 T + 779030 T^{2} - 268 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 72 T + 448286 T^{2} - 72 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 1626 T + 1498938 T^{2} + 1626 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.762122392317376711737093435320, −8.615669214236221593066922022917, −8.209871480198967937135438088196, −8.121561429497153745900538555480, −7.34968294503770647034917491364, −6.77783948051700661574075855378, −6.47980284614838081217995601922, −6.42446368842428570407377376871, −5.55631614844070404474073430616, −5.32566981380616519607906216617, −4.77312813825447432086487258664, −4.70077979524950616449964202921, −4.00410992233582864127978480211, −3.75602079898454941484652500482, −2.90997972760684705013016309415, −2.70466708524382448310719010827, −1.98413076513998612259315007903, −1.37338743986816212204338693117, −0.77772324666359004054623740249, −0.52704134419363556317130216397,
0.52704134419363556317130216397, 0.77772324666359004054623740249, 1.37338743986816212204338693117, 1.98413076513998612259315007903, 2.70466708524382448310719010827, 2.90997972760684705013016309415, 3.75602079898454941484652500482, 4.00410992233582864127978480211, 4.70077979524950616449964202921, 4.77312813825447432086487258664, 5.32566981380616519607906216617, 5.55631614844070404474073430616, 6.42446368842428570407377376871, 6.47980284614838081217995601922, 6.77783948051700661574075855378, 7.34968294503770647034917491364, 8.121561429497153745900538555480, 8.209871480198967937135438088196, 8.615669214236221593066922022917, 8.762122392317376711737093435320
