L(s) = 1 | − 2-s + 4-s − 5-s + 7-s − 8-s + 10-s + 11-s − 4.74·13-s − 14-s + 16-s − 4.74·17-s + 4.74·19-s − 20-s − 22-s − 4.74·23-s + 25-s + 4.74·26-s + 28-s + 2.74·29-s − 6.74·31-s − 32-s + 4.74·34-s − 35-s − 10.7·37-s − 4.74·38-s + 40-s + 4·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.377·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 1.31·13-s − 0.267·14-s + 0.250·16-s − 1.15·17-s + 1.08·19-s − 0.223·20-s − 0.213·22-s − 0.989·23-s + 0.200·25-s + 0.930·26-s + 0.188·28-s + 0.509·29-s − 1.21·31-s − 0.176·32-s + 0.813·34-s − 0.169·35-s − 1.76·37-s − 0.769·38-s + 0.158·40-s + 0.624·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9109323909\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9109323909\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
good | 13 | \( 1 + 4.74T + 13T^{2} \) |
| 17 | \( 1 + 4.74T + 17T^{2} \) |
| 19 | \( 1 - 4.74T + 19T^{2} \) |
| 23 | \( 1 + 4.74T + 23T^{2} \) |
| 29 | \( 1 - 2.74T + 29T^{2} \) |
| 31 | \( 1 + 6.74T + 31T^{2} \) |
| 37 | \( 1 + 10.7T + 37T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 - 6.74T + 47T^{2} \) |
| 53 | \( 1 - 1.25T + 53T^{2} \) |
| 59 | \( 1 + 2.74T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 - 4T + 71T^{2} \) |
| 73 | \( 1 + 0.744T + 73T^{2} \) |
| 79 | \( 1 + 4.74T + 79T^{2} \) |
| 83 | \( 1 + 8T + 83T^{2} \) |
| 89 | \( 1 - 7.48T + 89T^{2} \) |
| 97 | \( 1 + 5.25T + 97T^{2} \) |
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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
−7.892961283148553959244126926667, −7.29139060002777885667740653338, −6.92392220281245299589842728777, −5.88919343236635472487912923110, −5.14292359918261585565323250484, −4.36029373569550882188310309169, −3.53503284104572076127560360694, −2.50435420563622874003141740351, −1.80107209787269710138502592812, −0.52829117305309699357891062227,
0.52829117305309699357891062227, 1.80107209787269710138502592812, 2.50435420563622874003141740351, 3.53503284104572076127560360694, 4.36029373569550882188310309169, 5.14292359918261585565323250484, 5.88919343236635472487912923110, 6.92392220281245299589842728777, 7.29139060002777885667740653338, 7.892961283148553959244126926667