| L(s) = 1 | + 256·4-s − 6.07e3·7-s + 1.35e4·11-s + 4.91e4·16-s + 8.94e5·23-s + 1.38e6·25-s − 1.55e6·28-s − 3.17e5·29-s − 2.49e6·37-s + 9.18e6·43-s + 3.47e6·44-s + 1.89e7·49-s + 3.87e7·53-s + 8.38e6·64-s − 1.23e7·67-s − 6.21e7·71-s − 8.23e7·77-s + 2.48e7·79-s + 2.28e8·92-s + 3.55e8·100-s + 5.10e8·107-s + 1.36e8·109-s − 2.98e8·112-s − 2.41e8·113-s − 8.11e7·116-s − 6.33e8·121-s + 127-s + ⋯ |
| L(s) = 1 | + 4-s − 2.53·7-s + 0.926·11-s + 3/4·16-s + 3.19·23-s + 3.55·25-s − 2.53·28-s − 0.448·29-s − 1.33·37-s + 2.68·43-s + 0.926·44-s + 3.28·49-s + 4.90·53-s + 1/2·64-s − 0.611·67-s − 2.44·71-s − 2.34·77-s + 0.639·79-s + 3.19·92-s + 3.55·100-s + 3.89·107-s + 0.964·109-s − 1.89·112-s − 1.47·113-s − 0.448·116-s − 2.95·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(9-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{4} \cdot 3^{8} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+4)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(5.083655992\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.083655992\) |
| \(L(5)\) |
|
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 | $C_2$ | \( ( 1 - p^{7} T^{2} )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $D_{4}$ | \( 1 + 124 p^{2} T + 7494 p^{4} T^{2} + 124 p^{10} T^{3} + p^{16} T^{4} \) |
| good | 5 | $C_2^2 \wr C_2$ | \( 1 - 55588 p^{2} T^{2} + 783643216902 T^{4} - 55588 p^{18} T^{6} + p^{32} T^{8} \) |
| 11 | $D_{4}$ | \( ( 1 - 6780 T + 385462214 T^{2} - 6780 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 13 | $C_2^2 \wr C_2$ | \( 1 - 1612887940 T^{2} + 1300734062657673990 T^{4} - 1612887940 p^{16} T^{6} + p^{32} T^{8} \) |
| 17 | $C_2^2 \wr C_2$ | \( 1 - 16023366148 T^{2} + \)\(14\!\cdots\!66\)\( T^{4} - 16023366148 p^{16} T^{6} + p^{32} T^{8} \) |
| 19 | $C_2^2 \wr C_2$ | \( 1 + 26834594684 T^{2} + \)\(74\!\cdots\!38\)\( T^{4} + 26834594684 p^{16} T^{6} + p^{32} T^{8} \) |
| 23 | $D_{4}$ | \( ( 1 - 447036 T + 202011057158 T^{2} - 447036 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 29 | $D_{4}$ | \( ( 1 + 158532 T + 420149466566 T^{2} + 158532 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 31 | $C_2^2 \wr C_2$ | \( 1 - 1257474960388 T^{2} + \)\(17\!\cdots\!46\)\( T^{4} - 1257474960388 p^{16} T^{6} + p^{32} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 + 1247548 T + 7156715791686 T^{2} + 1247548 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 41 | $C_2^2 \wr C_2$ | \( 1 - 12045003373828 T^{2} + \)\(11\!\cdots\!06\)\( T^{4} - 12045003373828 p^{16} T^{6} + p^{32} T^{8} \) |
| 43 | $D_{4}$ | \( ( 1 - 4593284 T + 25927747162566 T^{2} - 4593284 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 47 | $C_2^2 \wr C_2$ | \( 1 - 29473105285636 T^{2} + \)\(52\!\cdots\!38\)\( T^{4} - 29473105285636 p^{16} T^{6} + p^{32} T^{8} \) |
| 53 | $D_{4}$ | \( ( 1 - 19363644 T + 215363679361094 T^{2} - 19363644 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 59 | $C_2^2 \wr C_2$ | \( 1 - 468430280695684 T^{2} + \)\(97\!\cdots\!46\)\( T^{4} - 468430280695684 p^{16} T^{6} + p^{32} T^{8} \) |
| 61 | $C_2^2 \wr C_2$ | \( 1 - 306906143267716 T^{2} + \)\(82\!\cdots\!18\)\( T^{4} - 306906143267716 p^{16} T^{6} + p^{32} T^{8} \) |
| 67 | $D_{4}$ | \( ( 1 + 6160124 T - 450645761195706 T^{2} + 6160124 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 71 | $D_{4}$ | \( ( 1 + 31084356 T + 692364955411334 T^{2} + 31084356 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 73 | $C_2^2 \wr C_2$ | \( 1 - 1018814792244484 T^{2} + \)\(36\!\cdots\!18\)\( T^{4} - 1018814792244484 p^{16} T^{6} + p^{32} T^{8} \) |
| 79 | $D_{4}$ | \( ( 1 - 12444868 T + 2983388778012678 T^{2} - 12444868 p^{8} T^{3} + p^{16} T^{4} )^{2} \) |
| 83 | $C_2^2 \wr C_2$ | \( 1 - 2088432522150532 T^{2} + \)\(90\!\cdots\!50\)\( T^{4} - 2088432522150532 p^{16} T^{6} + p^{32} T^{8} \) |
| 89 | $C_2^2 \wr C_2$ | \( 1 - 6583269594928900 T^{2} + \)\(30\!\cdots\!10\)\( T^{4} - 6583269594928900 p^{16} T^{6} + p^{32} T^{8} \) |
| 97 | $C_2^2 \wr C_2$ | \( 1 - 14985098969373700 T^{2} + \)\(11\!\cdots\!70\)\( T^{4} - 14985098969373700 p^{16} T^{6} + p^{32} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.661181704894215897438501436658, −7.63829128245688659412890420209, −7.30358752715334488630372528312, −7.24597884192721913566271057530, −7.05768724786029548583284372089, −6.93728383455566480077344277460, −6.47456274238237544974084451531, −6.25651530398311289931545028663, −6.04621305275000230480320739738, −5.71245098536291465361306857289, −5.21281419015383357950223686554, −4.87720460732729982594763290226, −4.86159010932274405797543178719, −3.92725749011122456150348248863, −3.84407122596527703585569298731, −3.62286779157695310083643901737, −3.05156713923510067185361190228, −2.80176656301739495150842915302, −2.65842387035695531675371997489, −2.51887217783058710612146326473, −1.73025026959653281406753281059, −1.06118226368688397455162326686, −0.963588243001840695215494223157, −0.884858493703040112864387384025, −0.26677516655495906949202094463,
0.26677516655495906949202094463, 0.884858493703040112864387384025, 0.963588243001840695215494223157, 1.06118226368688397455162326686, 1.73025026959653281406753281059, 2.51887217783058710612146326473, 2.65842387035695531675371997489, 2.80176656301739495150842915302, 3.05156713923510067185361190228, 3.62286779157695310083643901737, 3.84407122596527703585569298731, 3.92725749011122456150348248863, 4.86159010932274405797543178719, 4.87720460732729982594763290226, 5.21281419015383357950223686554, 5.71245098536291465361306857289, 6.04621305275000230480320739738, 6.25651530398311289931545028663, 6.47456274238237544974084451531, 6.93728383455566480077344277460, 7.05768724786029548583284372089, 7.24597884192721913566271057530, 7.30358752715334488630372528312, 7.63829128245688659412890420209, 8.661181704894215897438501436658
