| L(s) = 1 | + 5·3-s − 5·5-s − 30·7-s − 27·9-s − 71·11-s + 35·13-s − 25·15-s − 150·21-s − 5·23-s − 25·25-s − 260·27-s − 155·29-s + 88·31-s − 355·33-s + 150·35-s − 380·37-s + 175·39-s + 142·41-s + 155·43-s + 135·45-s − 455·47-s + 121·49-s + 275·53-s + 355·55-s + 873·59-s + 445·61-s + 810·63-s + ⋯ |
| L(s) = 1 | + 0.962·3-s − 0.447·5-s − 1.61·7-s − 9-s − 1.94·11-s + 0.746·13-s − 0.430·15-s − 1.55·21-s − 0.0453·23-s − 1/5·25-s − 1.85·27-s − 0.992·29-s + 0.509·31-s − 1.87·33-s + 0.724·35-s − 1.68·37-s + 0.718·39-s + 0.540·41-s + 0.549·43-s + 0.447·45-s − 1.41·47-s + 0.352·49-s + 0.712·53-s + 0.870·55-s + 1.92·59-s + 0.934·61-s + 1.61·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2085136 ^{s/2} \, \Gamma_{\C}(s+3/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.1582052701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1582052701\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 19 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - 5 T + 52 T^{2} - 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 5 | $D_{4}$ | \( 1 + p T + 2 p^{2} T^{2} + p^{4} T^{3} + p^{6} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + 30 T + 779 T^{2} + 30 p^{3} T^{3} + p^{6} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 71 T + 3716 T^{2} + 71 p^{3} T^{3} + p^{6} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 35 T + 3702 T^{2} - 35 p^{3} T^{3} + p^{6} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 9529 T^{2} + p^{6} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 5 T - 4378 T^{2} + 5 p^{3} T^{3} + p^{6} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 155 T + 19928 T^{2} + 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 31 | $D_{4}$ | \( 1 - 88 T + 8718 T^{2} - 88 p^{3} T^{3} + p^{6} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 380 T + 136878 T^{2} + 380 p^{3} T^{3} + p^{6} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 142 T + 135458 T^{2} - 142 p^{3} T^{3} + p^{6} T^{4} \) |
| 43 | $D_{4}$ | \( 1 - 155 T + 52086 T^{2} - 155 p^{3} T^{3} + p^{6} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 455 T + 255038 T^{2} + 455 p^{3} T^{3} + p^{6} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 275 T + 191846 T^{2} - 275 p^{3} T^{3} + p^{6} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 873 T + 591184 T^{2} - 873 p^{3} T^{3} + p^{6} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 445 T + 250812 T^{2} - 445 p^{3} T^{3} + p^{6} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 645 T + 674834 T^{2} + 645 p^{3} T^{3} + p^{6} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 1712 T + 1445258 T^{2} - 1712 p^{3} T^{3} + p^{6} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 990 T + 989267 T^{2} + 990 p^{3} T^{3} + p^{6} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 1274 T + 1391022 T^{2} - 1274 p^{3} T^{3} + p^{6} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 90 T + 1038382 T^{2} + 90 p^{3} T^{3} + p^{6} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 888 T + 1554274 T^{2} - 888 p^{3} T^{3} + p^{6} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 710 T + 220818 T^{2} + 710 p^{3} T^{3} + p^{6} T^{4} \) |
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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
−9.044677785179141828645965037369, −9.029167782124646585598271569683, −8.609456163258963838504126165994, −8.149269680125648424986675109164, −7.75964391961433642625284295767, −7.64863563215340404060239761439, −6.85400887412351911434309090761, −6.63801105897354697924467317540, −5.94835414791555536078988212849, −5.79519800647513560514533431636, −5.13128106272326251727369212943, −4.96883290582949944477452912692, −3.88039220595962100588609378968, −3.66625614898023446721353897152, −3.27278798802038271440682772352, −2.88889493157808403279340589222, −2.34834151893248409147297584307, −2.03418730314643836955228788121, −0.811800537377448312204552249538, −0.10393854461743198447016427216,
0.10393854461743198447016427216, 0.811800537377448312204552249538, 2.03418730314643836955228788121, 2.34834151893248409147297584307, 2.88889493157808403279340589222, 3.27278798802038271440682772352, 3.66625614898023446721353897152, 3.88039220595962100588609378968, 4.96883290582949944477452912692, 5.13128106272326251727369212943, 5.79519800647513560514533431636, 5.94835414791555536078988212849, 6.63801105897354697924467317540, 6.85400887412351911434309090761, 7.64863563215340404060239761439, 7.75964391961433642625284295767, 8.149269680125648424986675109164, 8.609456163258963838504126165994, 9.029167782124646585598271569683, 9.044677785179141828645965037369
