| L(s) = 1 | + 2·2-s − 3·3-s + 2·4-s + 5-s − 6·6-s + 4·8-s + 3·9-s + 2·10-s − 11-s − 6·12-s + 6·13-s − 3·15-s + 8·16-s + 3·17-s + 6·18-s − 6·19-s + 2·20-s − 2·22-s + 4·23-s − 12·24-s + 12·26-s − 2·29-s − 6·30-s − 6·31-s + 8·32-s + 3·33-s + 6·34-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 + 9-s + 0.632·10-s − 0.301·11-s − 1.73·12-s + 1.66·13-s − 0.774·15-s + 2·16-s + 0.727·17-s + 1.41·18-s − 1.37·19-s + 0.447·20-s − 0.426·22-s + 0.834·23-s − 2.44·24-s + 2.35·26-s − 0.371·29-s − 1.09·30-s − 1.07·31-s + 1.41·32-s + 0.522·33-s + 1.02·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 60025 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.975948723\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.975948723\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 - T + T^{2} \) |
| 7 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - p T + p T^{2} - p^{2} T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 + p T^{2} )( 1 + p T + p T^{2} ) \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 3 T - 8 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 + 6 T + 17 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 4 T - 7 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 6 T + 5 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 9 T + 34 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 10 T + 47 T^{2} - 10 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 6 T - 23 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 - 14 T + 129 T^{2} - 14 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 6 T - 37 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - T - 78 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 + 12 T + 55 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 15 T + p T^{2} )^{2} \) |
| show more | | |
| show less | | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.78094222734167145097970889775, −11.85785802397990959860986925353, −11.33942098867971228663494350102, −10.99737620983592802191619675271, −10.78968569566782096465472114887, −10.28502008616270416927402415308, −9.772519934337070000481049035500, −8.746758490679594185091921989594, −8.555596114853746859863822294233, −7.53664229933432265155589250927, −7.21578574839951255261214012418, −6.26072856732831130321761475364, −6.19023905252357600250940326177, −5.46763482707346474532401119803, −5.37048733688542037167117801113, −4.63393094357409494332828002858, −3.95921682451855141355502015075, −3.49522255371530849901832789870, −2.24029723271881156819273036130, −1.13303959293881987851224338427,
1.13303959293881987851224338427, 2.24029723271881156819273036130, 3.49522255371530849901832789870, 3.95921682451855141355502015075, 4.63393094357409494332828002858, 5.37048733688542037167117801113, 5.46763482707346474532401119803, 6.19023905252357600250940326177, 6.26072856732831130321761475364, 7.21578574839951255261214012418, 7.53664229933432265155589250927, 8.555596114853746859863822294233, 8.746758490679594185091921989594, 9.772519934337070000481049035500, 10.28502008616270416927402415308, 10.78968569566782096465472114887, 10.99737620983592802191619675271, 11.33942098867971228663494350102, 11.85785802397990959860986925353, 12.78094222734167145097970889775
