| L(s) = 1 | + 6·2-s + 13·4-s + 10·5-s − 12·8-s + 60·10-s − 28·11-s + 36·13-s − 147·16-s − 76·17-s − 160·19-s + 130·20-s − 168·22-s + 22·23-s + 75·25-s + 216·26-s + 250·29-s + 132·31-s − 366·32-s − 456·34-s − 416·37-s − 960·38-s − 120·40-s − 106·41-s − 666·43-s − 364·44-s + 132·46-s − 196·47-s + ⋯ |
| L(s) = 1 | + 2.12·2-s + 13/8·4-s + 0.894·5-s − 0.530·8-s + 1.89·10-s − 0.767·11-s + 0.768·13-s − 2.29·16-s − 1.08·17-s − 1.93·19-s + 1.45·20-s − 1.62·22-s + 0.199·23-s + 3/5·25-s + 1.62·26-s + 1.60·29-s + 0.764·31-s − 2.02·32-s − 2.30·34-s − 1.84·37-s − 4.09·38-s − 0.474·40-s − 0.403·41-s − 2.36·43-s − 1.24·44-s + 0.423·46-s − 0.608·47-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | $D_{4}$ | \( 1 - 3 p T + 23 T^{2} - 3 p^{4} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 28 T + 2658 T^{2} + 28 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 36 T + 4518 T^{2} - 36 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 76 T + 5438 T^{2} + 76 p^{3} T^{3} + p^{6} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + 160 T + 18766 T^{2} + 160 p^{3} T^{3} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 22 T - 2923 T^{2} - 22 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 250 T + 57203 T^{2} - 250 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 132 T + 43938 T^{2} - 132 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 416 T + 107578 T^{2} + 416 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 + 106 T + 138851 T^{2} + 106 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 666 T + 269853 T^{2} + 666 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 196 T + 209058 T^{2} + 196 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 952 T + 484002 T^{2} - 952 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 840 T + 445646 T^{2} - 840 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 98 T - 193437 T^{2} + 98 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 1286 T + 1005453 T^{2} + 1286 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 1064 T + 753846 T^{2} + 1064 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 172 T + 757582 T^{2} - 172 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 1240 T + 1278886 T^{2} + 1240 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 1906 T + 2051733 T^{2} + 1906 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 650 T + 1305611 T^{2} - 650 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 628 T + 1423942 T^{2} - 628 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−8.636507046398640773636141976387, −8.453565340003411716840278633476, −7.54886544649725310425440671097, −7.02359095070133088163426710018, −6.57393217042474465696054973490, −6.49268043773588287301138132624, −5.89444212664407642106808594454, −5.79163956650594377421196551775, −5.11673505555507812672986026592, −4.88877306990796603936891745585, −4.56161651225160412598496109888, −4.18632835847452375611412491895, −3.64759922069914234253102412208, −3.27974395025858111346575102147, −2.63423560755013275398142139247, −2.46860401114981664092656161682, −1.79423959066463284730612155503, −1.20295674488558415113867060658, 0, 0,
1.20295674488558415113867060658, 1.79423959066463284730612155503, 2.46860401114981664092656161682, 2.63423560755013275398142139247, 3.27974395025858111346575102147, 3.64759922069914234253102412208, 4.18632835847452375611412491895, 4.56161651225160412598496109888, 4.88877306990796603936891745585, 5.11673505555507812672986026592, 5.79163956650594377421196551775, 5.89444212664407642106808594454, 6.49268043773588287301138132624, 6.57393217042474465696054973490, 7.02359095070133088163426710018, 7.54886544649725310425440671097, 8.453565340003411716840278633476, 8.636507046398640773636141976387
