| L(s) = 1 | + 9·5-s + 5·7-s + 18·9-s − 3·11-s − 15·17-s − 38·19-s − 60·23-s + 25·25-s + 45·35-s + 85·43-s + 162·45-s − 75·47-s + 49·49-s − 27·55-s − 103·61-s + 90·63-s + 25·73-s − 15·77-s + 243·81-s + 180·83-s − 135·85-s − 342·95-s − 54·99-s − 204·101-s − 540·115-s − 75·119-s + 121·121-s + ⋯ |
| L(s) = 1 | + 9/5·5-s + 5/7·7-s + 2·9-s − 0.272·11-s − 0.882·17-s − 2·19-s − 2.60·23-s + 25-s + 9/7·35-s + 1.97·43-s + 18/5·45-s − 1.59·47-s + 49-s − 0.490·55-s − 1.68·61-s + 10/7·63-s + 0.342·73-s − 0.194·77-s + 3·81-s + 2.16·83-s − 1.58·85-s − 3.59·95-s − 0.545·99-s − 2.01·101-s − 4.69·115-s − 0.630·119-s + 121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.108644187\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.108644187\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| good | 3 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 5 | $C_2^2$ | \( 1 - 9 T + 56 T^{2} - 9 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 5 T - 24 T^{2} - 5 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 112 T^{2} + 3 p^{2} T^{3} + p^{4} T^{4} \) |
| 13 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 15 T - 64 T^{2} + 15 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 30 T + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 85 T + 5376 T^{2} - 85 p^{2} T^{3} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 75 T + 3416 T^{2} + 75 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 59 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 103 T + 6888 T^{2} + 103 p^{2} T^{3} + p^{4} T^{4} \) |
| 67 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 71 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 25 T - 4704 T^{2} - 25 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 90 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 97 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
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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
−14.31464744105884772672558991287, −13.98954748611141130826616464395, −13.25793806686218187541230197635, −13.22294055611218331123465894809, −12.40791984745472605653512357752, −12.08189190475909884171986166218, −10.83651742847128072456988721425, −10.66384767785405434824367159722, −10.04401244966865837397934554463, −9.623893377756596418000609728203, −9.062644504887732010909784046997, −8.145562520112669559470398627995, −7.66077486390326287623458516748, −6.63433232389462037367473591257, −6.27859806420955956445996812767, −5.56279149579478836720863030331, −4.44282070985266923527672563888, −4.19099285709592402724125876299, −2.01462065768240650349644573681, −1.94875373012814627765011922332,
1.94875373012814627765011922332, 2.01462065768240650349644573681, 4.19099285709592402724125876299, 4.44282070985266923527672563888, 5.56279149579478836720863030331, 6.27859806420955956445996812767, 6.63433232389462037367473591257, 7.66077486390326287623458516748, 8.145562520112669559470398627995, 9.062644504887732010909784046997, 9.623893377756596418000609728203, 10.04401244966865837397934554463, 10.66384767785405434824367159722, 10.83651742847128072456988721425, 12.08189190475909884171986166218, 12.40791984745472605653512357752, 13.22294055611218331123465894809, 13.25793806686218187541230197635, 13.98954748611141130826616464395, 14.31464744105884772672558991287
