L(s) = 1 | − 2·3-s − 4·5-s + 9-s + 4·11-s + 4·13-s + 8·15-s + 6·17-s − 2·19-s + 23-s + 11·25-s + 4·27-s + 6·29-s − 8·31-s − 8·33-s − 10·37-s − 8·39-s − 6·41-s + 12·43-s − 4·45-s − 8·47-s − 12·51-s + 2·53-s − 16·55-s + 4·57-s + 10·59-s − 4·61-s − 16·65-s + ⋯ |
L(s) = 1 | − 1.15·3-s − 1.78·5-s + 1/3·9-s + 1.20·11-s + 1.10·13-s + 2.06·15-s + 1.45·17-s − 0.458·19-s + 0.208·23-s + 11/5·25-s + 0.769·27-s + 1.11·29-s − 1.43·31-s − 1.39·33-s − 1.64·37-s − 1.28·39-s − 0.937·41-s + 1.82·43-s − 0.596·45-s − 1.16·47-s − 1.68·51-s + 0.274·53-s − 2.15·55-s + 0.529·57-s + 1.30·59-s − 0.512·61-s − 1.98·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72128 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.177875391\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.177875391\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 3 | \( 1 + 2 T + p T^{2} \) |
| 5 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.14094809084866, −13.70991724267572, −12.73654194529010, −12.38152800615763, −12.03270210872741, −11.69874901861136, −11.12053037291007, −10.81690098392200, −10.38891422665313, −9.544200818305378, −8.863160545576158, −8.455520398735653, −8.002449805534390, −7.313822943899434, −6.761593423787332, −6.475805820438188, −5.661623583575559, −5.252114699514158, −4.533340494118522, −3.966747153474501, −3.507934941229786, −3.099838437736743, −1.742211939471394, −0.9678603601404078, −0.5117363427292411,
0.5117363427292411, 0.9678603601404078, 1.742211939471394, 3.099838437736743, 3.507934941229786, 3.966747153474501, 4.533340494118522, 5.252114699514158, 5.661623583575559, 6.475805820438188, 6.761593423787332, 7.313822943899434, 8.002449805534390, 8.455520398735653, 8.863160545576158, 9.544200818305378, 10.38891422665313, 10.81690098392200, 11.12053037291007, 11.69874901861136, 12.03270210872741, 12.38152800615763, 12.73654194529010, 13.70991724267572, 14.14094809084866