| L(s) = 1 | + 7·2-s + 25·4-s − 10·5-s + 63·8-s − 70·10-s + 26·11-s − 14·13-s + 169·16-s + 16·17-s − 174·19-s − 250·20-s + 182·22-s − 184·23-s + 75·25-s − 98·26-s + 32·29-s − 330·31-s + 623·32-s + 112·34-s − 132·37-s − 1.21e3·38-s − 630·40-s + 200·41-s + 364·43-s + 650·44-s − 1.28e3·46-s + 292·47-s + ⋯ |
| L(s) = 1 | + 2.47·2-s + 25/8·4-s − 0.894·5-s + 2.78·8-s − 2.21·10-s + 0.712·11-s − 0.298·13-s + 2.64·16-s + 0.228·17-s − 2.10·19-s − 2.79·20-s + 1.76·22-s − 1.66·23-s + 3/5·25-s − 0.739·26-s + 0.204·29-s − 1.91·31-s + 3.44·32-s + 0.564·34-s − 0.586·37-s − 5.19·38-s − 2.49·40-s + 0.761·41-s + 1.29·43-s + 2.22·44-s − 4.12·46-s + 0.906·47-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
| \(L(2)\) |
\(\approx\) |
\(4.677357449\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.677357449\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 - 7 T + 3 p^{3} T^{2} - 7 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 26 T + 2406 T^{2} - 26 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 14 T + 2386 T^{2} + 14 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 16 T + 6558 T^{2} - 16 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 174 T + 18414 T^{2} + 174 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 8 p T + 19470 T^{2} + 8 p^{4} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 32 T + 19046 T^{2} - 32 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 330 T + 85430 T^{2} + 330 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 132 T + 103214 T^{2} + 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 200 T + 64542 T^{2} - 200 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 364 T + 172486 T^{2} - 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 - 292 T + 179390 T^{2} - 292 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 34 T + 103410 T^{2} + 34 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 364 T + 438374 T^{2} - 364 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 792 T + 340886 T^{2} + 792 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 788 T + 753430 T^{2} + 788 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 454 T + 304526 T^{2} + 454 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 778 T + 794698 T^{2} + 778 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 408 T + 994782 T^{2} - 408 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 1136 T + 1169990 T^{2} - 1136 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 36 T - 842170 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 498 T + 1796754 T^{2} - 498 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.887385867191743533014129876072, −8.356246765278081669153864259108, −7.914205089387325300040572876098, −7.65489329410655122500075766515, −7.17340154100322498602260083793, −6.81943569904632869238685319783, −6.23596249848109304354242060448, −6.07767907582692820432462592568, −5.59867279579813755536691947768, −5.36957035331700971908378455498, −4.56210057169494086381435836876, −4.41578748366982876425222906015, −4.04011758013969072746700681079, −3.95421370119213617280882138041, −3.26579639248389892643088366302, −3.00212000890085459665972406671, −2.16415724421380019292908478239, −1.97605024735257376920726237790, −1.15140610231543458891077733727, −0.27153326490052785236700977258,
0.27153326490052785236700977258, 1.15140610231543458891077733727, 1.97605024735257376920726237790, 2.16415724421380019292908478239, 3.00212000890085459665972406671, 3.26579639248389892643088366302, 3.95421370119213617280882138041, 4.04011758013969072746700681079, 4.41578748366982876425222906015, 4.56210057169494086381435836876, 5.36957035331700971908378455498, 5.59867279579813755536691947768, 6.07767907582692820432462592568, 6.23596249848109304354242060448, 6.81943569904632869238685319783, 7.17340154100322498602260083793, 7.65489329410655122500075766515, 7.914205089387325300040572876098, 8.356246765278081669153864259108, 8.887385867191743533014129876072
