L(s) = 1 | − 0.669·3-s − 5-s − 7-s − 2.55·9-s − 1.10·11-s − 4.93·13-s + 0.669·15-s − 7.84·17-s − 6.06·19-s + 0.669·21-s − 6.29·23-s + 25-s + 3.71·27-s − 6.73·29-s − 2.17·31-s + 0.742·33-s + 35-s − 0.816·37-s + 3.30·39-s + 3.61·41-s − 43-s + 2.55·45-s + 2.91·47-s + 49-s + 5.25·51-s + 0.776·53-s + 1.10·55-s + ⋯ |
L(s) = 1 | − 0.386·3-s − 0.447·5-s − 0.377·7-s − 0.850·9-s − 0.334·11-s − 1.36·13-s + 0.172·15-s − 1.90·17-s − 1.39·19-s + 0.146·21-s − 1.31·23-s + 0.200·25-s + 0.715·27-s − 1.25·29-s − 0.390·31-s + 0.129·33-s + 0.169·35-s − 0.134·37-s + 0.529·39-s + 0.565·41-s − 0.152·43-s + 0.380·45-s + 0.425·47-s + 0.142·49-s + 0.735·51-s + 0.106·53-s + 0.149·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.04569401030\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.04569401030\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 43 | \( 1 + T \) |
good | 3 | \( 1 + 0.669T + 3T^{2} \) |
| 11 | \( 1 + 1.10T + 11T^{2} \) |
| 13 | \( 1 + 4.93T + 13T^{2} \) |
| 17 | \( 1 + 7.84T + 17T^{2} \) |
| 19 | \( 1 + 6.06T + 19T^{2} \) |
| 23 | \( 1 + 6.29T + 23T^{2} \) |
| 29 | \( 1 + 6.73T + 29T^{2} \) |
| 31 | \( 1 + 2.17T + 31T^{2} \) |
| 37 | \( 1 + 0.816T + 37T^{2} \) |
| 41 | \( 1 - 3.61T + 41T^{2} \) |
| 47 | \( 1 - 2.91T + 47T^{2} \) |
| 53 | \( 1 - 0.776T + 53T^{2} \) |
| 59 | \( 1 + 3.37T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 7.64T + 67T^{2} \) |
| 71 | \( 1 - 0.259T + 71T^{2} \) |
| 73 | \( 1 + 11.0T + 73T^{2} \) |
| 79 | \( 1 - 9.33T + 79T^{2} \) |
| 83 | \( 1 + 10.4T + 83T^{2} \) |
| 89 | \( 1 + 0.837T + 89T^{2} \) |
| 97 | \( 1 + 3.83T + 97T^{2} \) |
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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
−8.112111082428128934030592701570, −7.29742789181408783019332336001, −6.65321090703745903201677353541, −5.98364467874608394115912337231, −5.22069643735287079696971480748, −4.41878204826386385200223425254, −3.80797284295607130083443936388, −2.55430257205687417735826256655, −2.14419126394437704523146276800, −0.10385772017134987363142510937,
0.10385772017134987363142510937, 2.14419126394437704523146276800, 2.55430257205687417735826256655, 3.80797284295607130083443936388, 4.41878204826386385200223425254, 5.22069643735287079696971480748, 5.98364467874608394115912337231, 6.65321090703745903201677353541, 7.29742789181408783019332336001, 8.112111082428128934030592701570