| L(s) = 1 | + 69·3-s + 155·5-s + 2.23e3·7-s + 621·9-s + 3.29e3·11-s − 1.34e4·13-s + 1.06e4·15-s − 3.22e4·17-s + 1.37e4·19-s + 1.54e5·21-s + 8.25e4·23-s + 1.38e4·25-s − 9.19e4·27-s − 1.27e4·29-s − 2.58e5·31-s + 2.27e5·33-s + 3.46e5·35-s − 1.49e5·37-s − 9.26e5·39-s + 3.39e5·41-s + 8.38e4·43-s + 9.62e4·45-s − 1.47e6·47-s + 2.23e6·49-s − 2.22e6·51-s − 9.45e5·53-s + 5.10e5·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s + 0.554·5-s + 2.46·7-s + 0.283·9-s + 0.746·11-s − 1.69·13-s + 0.818·15-s − 1.59·17-s + 0.458·19-s + 3.63·21-s + 1.41·23-s + 0.177·25-s − 0.898·27-s − 0.0970·29-s − 1.56·31-s + 1.10·33-s + 1.36·35-s − 0.484·37-s − 2.50·39-s + 0.768·41-s + 0.160·43-s + 0.157·45-s − 2.06·47-s + 2.71·49-s − 2.34·51-s − 0.872·53-s + 0.413·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+7/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(7.045835036\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.045835036\) |
| \(L(\frac{9}{2})\) |
|
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 | | \( 1 \) |
| 19 | $C_1$ | \( ( 1 - p^{3} T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 - 23 p T + 460 p^{2} T^{2} - 23 p^{8} T^{3} + p^{14} T^{4} \) |
| 5 | $D_{4}$ | \( 1 - 31 p T + 10178 T^{2} - 31 p^{8} T^{3} + p^{14} T^{4} \) |
| 7 | $D_{4}$ | \( 1 - 2238 T + 2775179 T^{2} - 2238 p^{7} T^{3} + p^{14} T^{4} \) |
| 11 | $D_{4}$ | \( 1 - 3295 T + 38080340 T^{2} - 3295 p^{7} T^{3} + p^{14} T^{4} \) |
| 13 | $D_{4}$ | \( 1 + 13427 T + 138664758 T^{2} + 13427 p^{7} T^{3} + p^{14} T^{4} \) |
| 17 | $D_{4}$ | \( 1 + 32256 T + 1073810905 T^{2} + 32256 p^{7} T^{3} + p^{14} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 82525 T + 5722043942 T^{2} - 82525 p^{7} T^{3} + p^{14} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12749 T + 39176144 p^{2} T^{2} + 12749 p^{7} T^{3} + p^{14} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 258944 T + 64961539758 T^{2} + 258944 p^{7} T^{3} + p^{14} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 149260 T + 183707435838 T^{2} + 149260 p^{7} T^{3} + p^{14} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 339130 T - 4364095006 T^{2} - 339130 p^{7} T^{3} + p^{14} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 83869 T + 6409629042 p T^{2} - 83869 p^{7} T^{3} + p^{14} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 1471025 T + 1488141383726 T^{2} + 1471025 p^{7} T^{3} + p^{14} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 945643 T + 2428814565902 T^{2} + 945643 p^{7} T^{3} + p^{14} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 969009 T + 3139777837024 T^{2} - 969009 p^{7} T^{3} + p^{14} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 1506755 T + 5925806441724 T^{2} + 1506755 p^{7} T^{3} + p^{14} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 1848219 T + 11014380276458 T^{2} - 1848219 p^{7} T^{3} + p^{14} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 3417184 T + 7686276501674 T^{2} - 3417184 p^{7} T^{3} + p^{14} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 2499822 T + 17574764549507 T^{2} + 2499822 p^{7} T^{3} + p^{14} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 2636926 T + 33245149452510 T^{2} + 2636926 p^{7} T^{3} + p^{14} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 10059354 T + 77358130536910 T^{2} - 10059354 p^{7} T^{3} + p^{14} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 3506160 T + 657141629522 p T^{2} + 3506160 p^{7} T^{3} + p^{14} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 60758 p T + 73244272821042 T^{2} - 60758 p^{8} T^{3} + p^{14} T^{4} \) |
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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
−10.90298893599899128511606111103, −10.22844225846040421958283151758, −9.435665292219585507758942729541, −9.202300552368330674348662596181, −8.958972265316424280701143677730, −8.361544269113017514538357046938, −7.86671574026825351601757398200, −7.73174629431931853038955644913, −6.87340921962216937674053426317, −6.71163363018131659653850396240, −5.41740733050567321746290065540, −5.33559623320169591781719374553, −4.52592911682541808389319615491, −4.45437369912080224557025067010, −3.37164009701460617257009795158, −2.88993383515185369297525507348, −2.05075396818652840545846865378, −2.02206856251423557377864514305, −1.43071383276805583748172620808, −0.48038851734849477595794653620,
0.48038851734849477595794653620, 1.43071383276805583748172620808, 2.02206856251423557377864514305, 2.05075396818652840545846865378, 2.88993383515185369297525507348, 3.37164009701460617257009795158, 4.45437369912080224557025067010, 4.52592911682541808389319615491, 5.33559623320169591781719374553, 5.41740733050567321746290065540, 6.71163363018131659653850396240, 6.87340921962216937674053426317, 7.73174629431931853038955644913, 7.86671574026825351601757398200, 8.361544269113017514538357046938, 8.958972265316424280701143677730, 9.202300552368330674348662596181, 9.435665292219585507758942729541, 10.22844225846040421958283151758, 10.90298893599899128511606111103
