L(s) = 1 | − 2-s − 2.35·3-s + 4-s + 2.35·6-s − 4.21·7-s − 8-s + 2.56·9-s + 3.59·11-s − 2.35·12-s − 0.585·13-s + 4.21·14-s + 16-s + 4.93·17-s − 2.56·18-s − 5.60·19-s + 9.94·21-s − 3.59·22-s + 6.62·23-s + 2.35·24-s + 0.585·26-s + 1.03·27-s − 4.21·28-s + 29-s − 5.81·31-s − 32-s − 8.48·33-s − 4.93·34-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.36·3-s + 0.5·4-s + 0.962·6-s − 1.59·7-s − 0.353·8-s + 0.854·9-s + 1.08·11-s − 0.680·12-s − 0.162·13-s + 1.12·14-s + 0.250·16-s + 1.19·17-s − 0.604·18-s − 1.28·19-s + 2.17·21-s − 0.767·22-s + 1.38·23-s + 0.481·24-s + 0.114·26-s + 0.198·27-s − 0.797·28-s + 0.185·29-s − 1.04·31-s − 0.176·32-s − 1.47·33-s − 0.846·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 29 | \( 1 - T \) |
good | 3 | \( 1 + 2.35T + 3T^{2} \) |
| 7 | \( 1 + 4.21T + 7T^{2} \) |
| 11 | \( 1 - 3.59T + 11T^{2} \) |
| 13 | \( 1 + 0.585T + 13T^{2} \) |
| 17 | \( 1 - 4.93T + 17T^{2} \) |
| 19 | \( 1 + 5.60T + 19T^{2} \) |
| 23 | \( 1 - 6.62T + 23T^{2} \) |
| 31 | \( 1 + 5.81T + 31T^{2} \) |
| 37 | \( 1 - 11.1T + 37T^{2} \) |
| 41 | \( 1 + 4.48T + 41T^{2} \) |
| 43 | \( 1 + 1.26T + 43T^{2} \) |
| 47 | \( 1 + 10.2T + 47T^{2} \) |
| 53 | \( 1 - 0.507T + 53T^{2} \) |
| 59 | \( 1 - 3.90T + 59T^{2} \) |
| 61 | \( 1 + 3.25T + 61T^{2} \) |
| 67 | \( 1 + 0.0605T + 67T^{2} \) |
| 71 | \( 1 - 10.0T + 71T^{2} \) |
| 73 | \( 1 + 7.57T + 73T^{2} \) |
| 79 | \( 1 - 5.73T + 79T^{2} \) |
| 83 | \( 1 + 15.4T + 83T^{2} \) |
| 89 | \( 1 - 0.111T + 89T^{2} \) |
| 97 | \( 1 - 12.3T + 97T^{2} \) |
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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
−9.406526298549816363954463086518, −8.458664006878440881028562778750, −7.21473643151257578329506854258, −6.54357117080564807104452933112, −6.15370931786913797220688280601, −5.20014141689749179406942779421, −3.92754314319804125134182924213, −2.90110586240431857775242070868, −1.18708198345144175297112436366, 0,
1.18708198345144175297112436366, 2.90110586240431857775242070868, 3.92754314319804125134182924213, 5.20014141689749179406942779421, 6.15370931786913797220688280601, 6.54357117080564807104452933112, 7.21473643151257578329506854258, 8.458664006878440881028562778750, 9.406526298549816363954463086518