| L(s) = 1 | + 2·2-s + 2·4-s + 8·7-s + 4·8-s + 3·9-s − 2·11-s − 2·13-s + 16·14-s + 8·16-s − 2·17-s + 6·18-s − 7·19-s − 4·22-s + 6·23-s − 4·26-s + 16·28-s − 9·29-s − 14·31-s + 8·32-s − 4·34-s + 6·36-s − 4·37-s − 14·38-s − 2·41-s + 2·43-s − 4·44-s + 12·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 3.02·7-s + 1.41·8-s + 9-s − 0.603·11-s − 0.554·13-s + 4.27·14-s + 2·16-s − 0.485·17-s + 1.41·18-s − 1.60·19-s − 0.852·22-s + 1.25·23-s − 0.784·26-s + 3.02·28-s − 1.67·29-s − 2.51·31-s + 1.41·32-s − 0.685·34-s + 36-s − 0.657·37-s − 2.27·38-s − 0.312·41-s + 0.304·43-s − 0.603·44-s + 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225625 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.612719751\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.612719751\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 5 | | \( 1 \) |
| 19 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 17 | $C_2^2$ | \( 1 + 2 T - 13 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 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 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2 T - 39 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 6 T - 11 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 4 T - 37 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 9 T + 22 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 10 T + 33 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + T - 70 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 17 T + p T^{2} )( 1 + 7 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + T - 78 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 11 T + 32 T^{2} - 11 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 6 T - 61 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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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
−11.21206814392667914991093789148, −10.90471578686655770881722212476, −10.47932195585320547667850945242, −10.47926428389554250498594186373, −9.231403414712642924893437783538, −9.066647111327061047407457202854, −8.284883554916736610418978676258, −7.935791464766070634966570650783, −7.43139367874779369906396395452, −7.28682255474782080712788413973, −6.70759651690475235308707442941, −5.53545734999154402714993023198, −5.47336379223912505359138887712, −4.80195433734292026316143436394, −4.75128138598538874450553548668, −3.98980113475805215523919362745, −3.85089679886022452512250861402, −2.45324272264237617901690965065, −1.77629153868512302240926579949, −1.60205579459502770893204042843,
1.60205579459502770893204042843, 1.77629153868512302240926579949, 2.45324272264237617901690965065, 3.85089679886022452512250861402, 3.98980113475805215523919362745, 4.75128138598538874450553548668, 4.80195433734292026316143436394, 5.47336379223912505359138887712, 5.53545734999154402714993023198, 6.70759651690475235308707442941, 7.28682255474782080712788413973, 7.43139367874779369906396395452, 7.935791464766070634966570650783, 8.284883554916736610418978676258, 9.066647111327061047407457202854, 9.231403414712642924893437783538, 10.47926428389554250498594186373, 10.47932195585320547667850945242, 10.90471578686655770881722212476, 11.21206814392667914991093789148
