| L(s) = 1 | + 3·3-s + 5-s − 4·7-s + 3·9-s − 11-s + 4·13-s + 3·15-s − 3·17-s + 5·19-s − 12·21-s − 3·23-s + 5·25-s − 12·29-s − 31-s − 3·33-s − 4·35-s + 5·37-s + 12·39-s − 20·41-s + 8·43-s + 3·45-s + 47-s + 9·49-s − 9·51-s + 9·53-s − 55-s + 15·57-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s − 1.51·7-s + 9-s − 0.301·11-s + 1.10·13-s + 0.774·15-s − 0.727·17-s + 1.14·19-s − 2.61·21-s − 0.625·23-s + 25-s − 2.22·29-s − 0.179·31-s − 0.522·33-s − 0.676·35-s + 0.821·37-s + 1.92·39-s − 3.12·41-s + 1.21·43-s + 0.447·45-s + 0.145·47-s + 9/7·49-s − 1.26·51-s + 1.23·53-s − 0.134·55-s + 1.98·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 12544 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.586358792\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.586358792\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + 4 T + p T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - T - 4 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + T - 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 + 3 T - 8 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 5 T + 6 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2^2$ | \( 1 + 3 T - 14 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + T - 30 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 5 T - 12 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 9 T + 28 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 3 T - 50 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2^2$ | \( 1 + 3 T - 52 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 67 | $C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 71 | $C_2$ | \( ( 1 + 16 T + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 + 17 T + p T^{2} ) \) |
| 79 | $C_2^2$ | \( 1 + 11 T + 42 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 9 T - 8 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
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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
−13.70305202555260878798490505558, −13.34836502465437859034610198566, −13.24688432492786032185953258752, −12.68548751411138965689879934653, −11.80082940926999881885742185286, −11.36073117391426913042391564366, −10.40217630212134984427352541425, −10.16813813781579635199109532636, −9.412728255827074359655560748809, −8.901421381993921158664686369373, −8.869829507626480074278667273485, −8.015935956480590823106124493460, −7.34359416371749824397387048395, −6.79382649383715623852401979701, −5.98782144721457735788228088405, −5.47814205782257414762550558126, −4.17284991819335405692382112761, −3.33607576344146875796464983280, −3.06863409097450051281485096062, −2.01435326820631659756192230674,
2.01435326820631659756192230674, 3.06863409097450051281485096062, 3.33607576344146875796464983280, 4.17284991819335405692382112761, 5.47814205782257414762550558126, 5.98782144721457735788228088405, 6.79382649383715623852401979701, 7.34359416371749824397387048395, 8.015935956480590823106124493460, 8.869829507626480074278667273485, 8.901421381993921158664686369373, 9.412728255827074359655560748809, 10.16813813781579635199109532636, 10.40217630212134984427352541425, 11.36073117391426913042391564366, 11.80082940926999881885742185286, 12.68548751411138965689879934653, 13.24688432492786032185953258752, 13.34836502465437859034610198566, 13.70305202555260878798490505558
