L(s) = 1 | − 2-s + 4-s − 3.46·5-s − 3.72·7-s − 8-s + 3.46·10-s + 5.41·11-s + 5.45·13-s + 3.72·14-s + 16-s + 5.94·17-s + 5.41·19-s − 3.46·20-s − 5.41·22-s + 4.72·23-s + 6.99·25-s − 5.45·26-s − 3.72·28-s + 5.68·29-s + 4.89·31-s − 32-s − 5.94·34-s + 12.8·35-s − 10.8·37-s − 5.41·38-s + 3.46·40-s + 10.6·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.54·5-s − 1.40·7-s − 0.353·8-s + 1.09·10-s + 1.63·11-s + 1.51·13-s + 0.995·14-s + 0.250·16-s + 1.44·17-s + 1.24·19-s − 0.774·20-s − 1.15·22-s + 0.984·23-s + 1.39·25-s − 1.06·26-s − 0.703·28-s + 1.05·29-s + 0.879·31-s − 0.176·32-s − 1.01·34-s + 2.18·35-s − 1.77·37-s − 0.877·38-s + 0.547·40-s + 1.67·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.277566282\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.277566282\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
good | 5 | \( 1 + 3.46T + 5T^{2} \) |
| 7 | \( 1 + 3.72T + 7T^{2} \) |
| 11 | \( 1 - 5.41T + 11T^{2} \) |
| 13 | \( 1 - 5.45T + 13T^{2} \) |
| 17 | \( 1 - 5.94T + 17T^{2} \) |
| 19 | \( 1 - 5.41T + 19T^{2} \) |
| 23 | \( 1 - 4.72T + 23T^{2} \) |
| 29 | \( 1 - 5.68T + 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 + 10.8T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 12.8T + 43T^{2} \) |
| 47 | \( 1 - 8.22T + 47T^{2} \) |
| 53 | \( 1 + 11.8T + 53T^{2} \) |
| 59 | \( 1 - 6.31T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 + 8.30T + 67T^{2} \) |
| 71 | \( 1 - 0.883T + 71T^{2} \) |
| 73 | \( 1 - 6.87T + 73T^{2} \) |
| 79 | \( 1 - 2.27T + 79T^{2} \) |
| 83 | \( 1 - 4.53T + 83T^{2} \) |
| 89 | \( 1 - 1.46T + 89T^{2} \) |
| 97 | \( 1 - 18.1T + 97T^{2} \) |
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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
−7.84427222389221767066956385238, −7.17525018590003544572445725269, −6.58929558871871113623198910978, −6.11834415596617962045561596984, −5.02864319170753987946318460388, −3.83300415780938243951518195097, −3.49462149631473117949783745480, −3.04863793083016725872158934397, −1.21240777037673315006809149381, −0.76619084362714547969261330820,
0.76619084362714547969261330820, 1.21240777037673315006809149381, 3.04863793083016725872158934397, 3.49462149631473117949783745480, 3.83300415780938243951518195097, 5.02864319170753987946318460388, 6.11834415596617962045561596984, 6.58929558871871113623198910978, 7.17525018590003544572445725269, 7.84427222389221767066956385238