| L(s) = 1 | − 2·2-s + 3·4-s + 3·5-s + 5·7-s − 4·8-s − 6·10-s − 3·11-s − 2·13-s − 10·14-s + 5·16-s + 6·17-s − 2·19-s + 9·20-s + 6·22-s − 6·23-s + 5·25-s + 4·26-s + 15·28-s + 9·29-s − 14·31-s − 6·32-s − 12·34-s + 15·35-s + 10·37-s + 4·38-s − 12·40-s + 4·43-s + ⋯ |
| L(s) = 1 | − 1.41·2-s + 3/2·4-s + 1.34·5-s + 1.88·7-s − 1.41·8-s − 1.89·10-s − 0.904·11-s − 0.554·13-s − 2.67·14-s + 5/4·16-s + 1.45·17-s − 0.458·19-s + 2.01·20-s + 1.27·22-s − 1.25·23-s + 25-s + 0.784·26-s + 2.83·28-s + 1.67·29-s − 2.51·31-s − 1.06·32-s − 2.05·34-s + 2.53·35-s + 1.64·37-s + 0.648·38-s − 1.89·40-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1285956 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.690662146\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.690662146\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 3 | | \( 1 \) |
| 7 | $C_2$ | \( 1 - 5 T + p T^{2} \) |
| good | 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 - 6 T + 19 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 2 T - 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 6 T + 13 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 11 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 2 T - 69 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 9 T - 2 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 - 13 T + 72 T^{2} - 13 p T^{3} + p^{2} T^{4} \) |
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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
−10.26980607198451944229242033133, −9.569685192289642502328349543342, −9.332961177908088222805515235009, −8.775262187367468271700633997296, −8.268844182903131002529243359122, −8.088168891708433689094398377918, −7.62184572398253721840102770906, −7.50517691809524325768051837531, −6.80973395041526012848909616557, −6.07009735323186838577893313892, −6.05326367348329007999016342382, −5.37767761339440120497449487034, −4.95863252746363072736864922124, −4.69349823285998869070588357282, −3.73695121283881014905113206878, −3.02531344070315714082326889820, −2.39150605009662662378889033192, −1.85763538174346659118100126300, −1.66039774469326545763691846884, −0.72350189192768568880147182121,
0.72350189192768568880147182121, 1.66039774469326545763691846884, 1.85763538174346659118100126300, 2.39150605009662662378889033192, 3.02531344070315714082326889820, 3.73695121283881014905113206878, 4.69349823285998869070588357282, 4.95863252746363072736864922124, 5.37767761339440120497449487034, 6.05326367348329007999016342382, 6.07009735323186838577893313892, 6.80973395041526012848909616557, 7.50517691809524325768051837531, 7.62184572398253721840102770906, 8.088168891708433689094398377918, 8.268844182903131002529243359122, 8.775262187367468271700633997296, 9.332961177908088222805515235009, 9.569685192289642502328349543342, 10.26980607198451944229242033133
