| L(s) = 1 | − 2-s − 3-s + 6·5-s + 6-s − 7-s + 8-s + 3·9-s − 6·10-s + 6·11-s + 14-s − 6·15-s − 16-s + 3·17-s − 3·18-s + 2·19-s + 21-s − 6·22-s − 24-s + 17·25-s − 8·27-s − 6·29-s + 6·30-s + 8·31-s − 6·33-s − 3·34-s − 6·35-s − 7·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 2.68·5-s + 0.408·6-s − 0.377·7-s + 0.353·8-s + 9-s − 1.89·10-s + 1.80·11-s + 0.267·14-s − 1.54·15-s − 1/4·16-s + 0.727·17-s − 0.707·18-s + 0.458·19-s + 0.218·21-s − 1.27·22-s − 0.204·24-s + 17/5·25-s − 1.53·27-s − 1.11·29-s + 1.09·30-s + 1.43·31-s − 1.04·33-s − 0.514·34-s − 1.01·35-s − 1.15·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 114244 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 114244 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.656258611\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.656258611\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T + T^{2} \) |
| 13 | | \( 1 \) |
| good | 3 | $C_2^2$ | \( 1 + T - 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 - 6 T + 25 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 2 T - 15 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 + 6 T + 7 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 7 T + 12 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - T - 42 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 6 T - 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 8 T + 3 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 3 T - 62 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 6 T - 53 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 10 T + 3 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−11.60967619699366416382551611675, −11.35543051478638778090258748762, −10.49186441281234235838314060196, −10.19873472295498509785689001808, −9.864935025021312356477139897005, −9.572584113146321138249905525998, −9.101405691148172230036869748750, −9.001602777616016471459907645130, −8.026027357253440522006233339658, −7.42364860981592227254693409780, −6.63631819452094475795831718184, −6.62572093069655918811370948820, −5.92134663573667486756704022253, −5.59186003148210604003732412198, −5.06130526409527960770800136433, −4.20393011018552674614268859520, −3.55887349022475377528249598166, −2.46865455591262811545435441123, −1.51527637428858548459108106953, −1.39614105227617374038853612553,
1.39614105227617374038853612553, 1.51527637428858548459108106953, 2.46865455591262811545435441123, 3.55887349022475377528249598166, 4.20393011018552674614268859520, 5.06130526409527960770800136433, 5.59186003148210604003732412198, 5.92134663573667486756704022253, 6.62572093069655918811370948820, 6.63631819452094475795831718184, 7.42364860981592227254693409780, 8.026027357253440522006233339658, 9.001602777616016471459907645130, 9.101405691148172230036869748750, 9.572584113146321138249905525998, 9.864935025021312356477139897005, 10.19873472295498509785689001808, 10.49186441281234235838314060196, 11.35543051478638778090258748762, 11.60967619699366416382551611675
