| L(s) = 1 | + 2·2-s − 3·3-s + 2·4-s + 5-s − 6·6-s + 4·8-s + 6·9-s + 2·10-s − 4·11-s − 6·12-s + 13-s − 3·15-s + 8·16-s + 14·17-s + 12·18-s + 8·19-s + 2·20-s − 8·22-s + 3·23-s − 12·24-s + 2·26-s − 9·27-s + 10·29-s − 6·30-s + 6·31-s + 8·32-s + 12·33-s + ⋯ |
| L(s) = 1 | + 1.41·2-s − 1.73·3-s + 4-s + 0.447·5-s − 2.44·6-s + 1.41·8-s + 2·9-s + 0.632·10-s − 1.20·11-s − 1.73·12-s + 0.277·13-s − 0.774·15-s + 2·16-s + 3.39·17-s + 2.82·18-s + 1.83·19-s + 0.447·20-s − 1.70·22-s + 0.625·23-s − 2.44·24-s + 0.392·26-s − 1.73·27-s + 1.85·29-s − 1.09·30-s + 1.07·31-s + 1.41·32-s + 2.08·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 342225 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.210593074\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.210593074\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 3 | $C_2$ | \( 1 + p T + p T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 13 | $C_2$ | \( 1 - T + T^{2} \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 4 T + 5 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 3 T - 14 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 10 T + 71 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 - 6 T + 5 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 + 2 T - 37 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + T - 42 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 2 T - 43 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 10 T + 41 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - 5 T - 36 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 12 T + 77 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
| 79 | $C_2^2$ | \( 1 + 9 T + 2 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2^2$ | \( 1 + 6 T - 47 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 8 T - 33 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
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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
−10.76656512357368015789789075764, −10.61887046484513106475598827910, −10.14001190949118419085740026825, −10.06129530140312149207053983849, −9.579458359839805378285795795471, −8.587680527767102316390569294290, −7.83115393055015799118969161081, −7.81482706316337914993432126921, −7.05240699458611398985262140488, −6.85495776109093851411564861759, −6.04425277140667517137271145754, −5.52603937720763000826171291697, −5.44990751480065995910414462417, −4.94048894625964911573209724862, −4.85839083876370046766625673965, −3.83648819449806990970902039682, −3.29107124636205639494595617081, −2.87830664405620156538570882640, −1.36463785067428000044570072520, −1.14399414847967889280047753125,
1.14399414847967889280047753125, 1.36463785067428000044570072520, 2.87830664405620156538570882640, 3.29107124636205639494595617081, 3.83648819449806990970902039682, 4.85839083876370046766625673965, 4.94048894625964911573209724862, 5.44990751480065995910414462417, 5.52603937720763000826171291697, 6.04425277140667517137271145754, 6.85495776109093851411564861759, 7.05240699458611398985262140488, 7.81482706316337914993432126921, 7.83115393055015799118969161081, 8.587680527767102316390569294290, 9.579458359839805378285795795471, 10.06129530140312149207053983849, 10.14001190949118419085740026825, 10.61887046484513106475598827910, 10.76656512357368015789789075764
