L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 4·7-s + 8-s + 9-s + 2·12-s − 2·13-s + 4·14-s + 16-s + 18-s + 5·19-s + 8·21-s − 3·23-s + 2·24-s − 2·26-s − 4·27-s + 4·28-s + 6·29-s − 4·31-s + 32-s + 36-s − 37-s + 5·38-s − 4·39-s − 9·41-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 1.51·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s + 0.235·18-s + 1.14·19-s + 1.74·21-s − 0.625·23-s + 0.408·24-s − 0.392·26-s − 0.769·27-s + 0.755·28-s + 1.11·29-s − 0.718·31-s + 0.176·32-s + 1/6·36-s − 0.164·37-s + 0.811·38-s − 0.640·39-s − 1.40·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1850 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.610082949\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.610082949\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 37 | \( 1 + T \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 7 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−9.072173867371578628731053101358, −8.325625935227627786598543670323, −7.72079969937173604461699084525, −7.13119834716081850869554973881, −5.81720251329883585877796210753, −5.03810072824874883598271911692, −4.28076506479936391749067520638, −3.28985181298884438738972774519, −2.41270668373056314614641267397, −1.51220124560110813647901832471,
1.51220124560110813647901832471, 2.41270668373056314614641267397, 3.28985181298884438738972774519, 4.28076506479936391749067520638, 5.03810072824874883598271911692, 5.81720251329883585877796210753, 7.13119834716081850869554973881, 7.72079969937173604461699084525, 8.325625935227627786598543670323, 9.072173867371578628731053101358