| L(s) = 1 | − 3·3-s − 7-s + 2·9-s − 2·11-s + 8·13-s + 17-s + 3·21-s − 3·23-s + 6·27-s − 5·29-s + 6·31-s + 6·33-s + 16·37-s − 24·39-s − 6·41-s + 12·43-s − 6·47-s − 2·49-s − 3·51-s + 3·53-s − 10·59-s − 11·61-s − 2·63-s − 16·67-s + 9·69-s + 6·71-s + 23·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 0.377·7-s + 2/3·9-s − 0.603·11-s + 2.21·13-s + 0.242·17-s + 0.654·21-s − 0.625·23-s + 1.15·27-s − 0.928·29-s + 1.07·31-s + 1.04·33-s + 2.63·37-s − 3.84·39-s − 0.937·41-s + 1.82·43-s − 0.875·47-s − 2/7·49-s − 0.420·51-s + 0.412·53-s − 1.30·59-s − 1.40·61-s − 0.251·63-s − 1.95·67-s + 1.08·69-s + 0.712·71-s + 2.69·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.076323100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.076323100\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + p T + 7 T^{2} + p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 3 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 8 T + 37 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - T + 33 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 3 T + 17 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 5 T + 63 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 16 T + 133 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 47 | $D_{4}$ | \( 1 + 6 T + 23 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 3 T + 47 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 + 10 T + 123 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 + 11 T + 121 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 71 | $C_4$ | \( 1 - 6 T + 26 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 - 23 T + 277 T^{2} - 23 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 5 T + 153 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 27 T + 337 T^{2} - 27 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 25 T + 303 T^{2} + 25 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - T + 193 T^{2} - p T^{3} + p^{2} T^{4} \) |
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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
−8.302237848030734480471528634208, −8.234606243764627901304360733427, −7.76309679698595714328795093651, −7.60764956526130263936496252488, −6.73593281513579639663115836049, −6.64431581847945195977035837931, −6.10260919981987429322664830186, −6.03837181311176266923668753874, −5.76369652525753023369117691543, −5.42550118355934300981434878868, −4.85519427458334599103766229175, −4.54559173449320739024063198172, −4.06067590764984919338574741528, −3.71385126388493105999030899121, −2.96302526302193951353364385140, −2.96234805469970261260666499545, −2.09668634322180872240734963371, −1.46934169082693586067313946329, −0.862705828975303626260603419420, −0.44401108391812806689103824226,
0.44401108391812806689103824226, 0.862705828975303626260603419420, 1.46934169082693586067313946329, 2.09668634322180872240734963371, 2.96234805469970261260666499545, 2.96302526302193951353364385140, 3.71385126388493105999030899121, 4.06067590764984919338574741528, 4.54559173449320739024063198172, 4.85519427458334599103766229175, 5.42550118355934300981434878868, 5.76369652525753023369117691543, 6.03837181311176266923668753874, 6.10260919981987429322664830186, 6.64431581847945195977035837931, 6.73593281513579639663115836049, 7.60764956526130263936496252488, 7.76309679698595714328795093651, 8.234606243764627901304360733427, 8.302237848030734480471528634208
