| L(s) = 1 | + 2-s + 3-s + 5-s + 6-s − 8-s + 10-s − 6·11-s − 12·13-s + 15-s − 16-s − 4·19-s − 6·22-s − 24-s − 12·26-s − 27-s − 16·29-s + 30-s + 2·31-s − 6·33-s − 4·37-s − 4·38-s − 12·39-s − 40-s + 20·41-s − 12·43-s − 2·47-s − 48-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.447·5-s + 0.408·6-s − 0.353·8-s + 0.316·10-s − 1.80·11-s − 3.32·13-s + 0.258·15-s − 1/4·16-s − 0.917·19-s − 1.27·22-s − 0.204·24-s − 2.35·26-s − 0.192·27-s − 2.97·29-s + 0.182·30-s + 0.359·31-s − 1.04·33-s − 0.657·37-s − 0.648·38-s − 1.92·39-s − 0.158·40-s + 3.12·41-s − 1.82·43-s − 0.291·47-s − 0.144·48-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2160900 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4258948196\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4258948196\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 3 | $C_2$ | \( 1 - T + T^{2} \) |
| 5 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 11 | $C_2^2$ | \( 1 + 6 T + 25 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 4 T - 3 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 + 4 T - 21 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 + 2 T - 43 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 10 T + 47 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 13 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 14 T + 129 T^{2} + 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 6 T - 37 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 8 T - 15 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 235 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
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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
−9.733171635385472453492115096318, −9.377071461683216012937606767413, −9.167320926604705985165872916313, −8.319130356596470936114698004341, −7.966088386936715335653715951346, −7.79520365894919479570142299135, −7.23054771692644943375253196648, −7.08620738631368080404293649529, −6.40852104509036981222482154269, −5.82665282412317686706775607810, −5.40046962788894292536140222082, −4.98864715133047891758936649047, −4.96895132438605431818564831077, −4.20273572760082607728198665256, −3.80132773337023542785425604962, −2.95878815223567889306776663349, −2.71432693468256038752239208239, −2.14789968791979629729907490652, −1.98756276704077858380754936997, −0.19763365948432917107394792803,
0.19763365948432917107394792803, 1.98756276704077858380754936997, 2.14789968791979629729907490652, 2.71432693468256038752239208239, 2.95878815223567889306776663349, 3.80132773337023542785425604962, 4.20273572760082607728198665256, 4.96895132438605431818564831077, 4.98864715133047891758936649047, 5.40046962788894292536140222082, 5.82665282412317686706775607810, 6.40852104509036981222482154269, 7.08620738631368080404293649529, 7.23054771692644943375253196648, 7.79520365894919479570142299135, 7.966088386936715335653715951346, 8.319130356596470936114698004341, 9.167320926604705985165872916313, 9.377071461683216012937606767413, 9.733171635385472453492115096318
