L(s) = 1 | + 2-s + 2·3-s + 4-s − 5-s + 2·6-s + 8-s + 9-s − 10-s − 4·11-s + 2·12-s + 13-s − 2·15-s + 16-s + 18-s + 6·19-s − 20-s − 4·22-s − 2·23-s + 2·24-s + 25-s + 26-s − 4·27-s + 6·29-s − 2·30-s + 8·31-s + 32-s − 8·33-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.447·5-s + 0.816·6-s + 0.353·8-s + 1/3·9-s − 0.316·10-s − 1.20·11-s + 0.577·12-s + 0.277·13-s − 0.516·15-s + 1/4·16-s + 0.235·18-s + 1.37·19-s − 0.223·20-s − 0.852·22-s − 0.417·23-s + 0.408·24-s + 1/5·25-s + 0.196·26-s − 0.769·27-s + 1.11·29-s − 0.365·30-s + 1.43·31-s + 0.176·32-s − 1.39·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6370 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6370 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.402352483\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.402352483\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 13 | \( 1 - T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 8 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 16 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−8.007054160865378911540908229557, −7.48061510279841436780155007185, −6.72108649576583257192871175852, −5.72015307856771059831749227507, −5.14537333567057773295970742851, −4.23544530913740919414019244874, −3.54746817566416023123137896609, −2.77735028701165982264444897965, −2.36359902645512386425637529923, −0.929045819064038267332735462655,
0.929045819064038267332735462655, 2.36359902645512386425637529923, 2.77735028701165982264444897965, 3.54746817566416023123137896609, 4.23544530913740919414019244874, 5.14537333567057773295970742851, 5.72015307856771059831749227507, 6.72108649576583257192871175852, 7.48061510279841436780155007185, 8.007054160865378911540908229557