L(s) = 1 | + 2-s − 1.77·3-s + 4-s − 1.77·6-s − 2.69·7-s + 8-s + 0.144·9-s + 5.54·11-s − 1.77·12-s − 2.91·13-s − 2.69·14-s + 16-s + 4.91·17-s + 0.144·18-s + 19-s + 4.77·21-s + 5.54·22-s + 3.60·23-s − 1.77·24-s − 2.91·26-s + 5.06·27-s − 2.69·28-s + 1.08·29-s + 7.54·31-s + 32-s − 9.83·33-s + 4.91·34-s + ⋯ |
L(s) = 1 | + 0.707·2-s − 1.02·3-s + 0.5·4-s − 0.723·6-s − 1.01·7-s + 0.353·8-s + 0.0483·9-s + 1.67·11-s − 0.511·12-s − 0.809·13-s − 0.719·14-s + 0.250·16-s + 1.19·17-s + 0.0341·18-s + 0.229·19-s + 1.04·21-s + 1.18·22-s + 0.752·23-s − 0.361·24-s − 0.572·26-s + 0.974·27-s − 0.508·28-s + 0.200·29-s + 1.35·31-s + 0.176·32-s − 1.71·33-s + 0.843·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.639330195\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.639330195\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
good | 3 | \( 1 + 1.77T + 3T^{2} \) |
| 7 | \( 1 + 2.69T + 7T^{2} \) |
| 11 | \( 1 - 5.54T + 11T^{2} \) |
| 13 | \( 1 + 2.91T + 13T^{2} \) |
| 17 | \( 1 - 4.91T + 17T^{2} \) |
| 23 | \( 1 - 3.60T + 23T^{2} \) |
| 29 | \( 1 - 1.08T + 29T^{2} \) |
| 31 | \( 1 - 7.54T + 31T^{2} \) |
| 37 | \( 1 - 4.54T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 9.54T + 43T^{2} \) |
| 47 | \( 1 + 0.836T + 47T^{2} \) |
| 53 | \( 1 - 9.78T + 53T^{2} \) |
| 59 | \( 1 - 12.9T + 59T^{2} \) |
| 61 | \( 1 + 7.38T + 61T^{2} \) |
| 67 | \( 1 - 2.85T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 - 5.15T + 73T^{2} \) |
| 79 | \( 1 + 3.09T + 79T^{2} \) |
| 83 | \( 1 - 1.71T + 83T^{2} \) |
| 89 | \( 1 + 5.09T + 89T^{2} \) |
| 97 | \( 1 + 17.2T + 97T^{2} \) |
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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
−10.02137410483440977469406595274, −9.548316411523299794624782733183, −8.313911214172205567137647701297, −6.94564488845183948831283162403, −6.56949817105273219663539006717, −5.71920584680150902339911179367, −4.89805723004525089290368217354, −3.79832783309507638366050963113, −2.84385201194011928051731325330, −0.992063788575269969793546874418,
0.992063788575269969793546874418, 2.84385201194011928051731325330, 3.79832783309507638366050963113, 4.89805723004525089290368217354, 5.71920584680150902339911179367, 6.56949817105273219663539006717, 6.94564488845183948831283162403, 8.313911214172205567137647701297, 9.548316411523299794624782733183, 10.02137410483440977469406595274