| L(s) = 1 | + 2-s + 2.44·3-s + 4-s − 5-s + 2.44·6-s − 1.84·7-s + 8-s + 2.97·9-s − 10-s − 6.51·11-s + 2.44·12-s + 6.36·13-s − 1.84·14-s − 2.44·15-s + 16-s + 7.68·17-s + 2.97·18-s + 1.62·19-s − 20-s − 4.49·21-s − 6.51·22-s − 2.90·23-s + 2.44·24-s + 25-s + 6.36·26-s − 0.0646·27-s − 1.84·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.41·3-s + 0.5·4-s − 0.447·5-s + 0.997·6-s − 0.695·7-s + 0.353·8-s + 0.991·9-s − 0.316·10-s − 1.96·11-s + 0.705·12-s + 1.76·13-s − 0.491·14-s − 0.631·15-s + 0.250·16-s + 1.86·17-s + 0.700·18-s + 0.372·19-s − 0.223·20-s − 0.981·21-s − 1.38·22-s − 0.605·23-s + 0.498·24-s + 0.200·25-s + 1.24·26-s − 0.0124·27-s − 0.347·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.758399687\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.758399687\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 + 1.84T + 7T^{2} \) |
| 11 | \( 1 + 6.51T + 11T^{2} \) |
| 13 | \( 1 - 6.36T + 13T^{2} \) |
| 17 | \( 1 - 7.68T + 17T^{2} \) |
| 19 | \( 1 - 1.62T + 19T^{2} \) |
| 23 | \( 1 + 2.90T + 23T^{2} \) |
| 29 | \( 1 - 10.1T + 29T^{2} \) |
| 31 | \( 1 - 0.289T + 31T^{2} \) |
| 37 | \( 1 + 0.634T + 37T^{2} \) |
| 41 | \( 1 + 4.22T + 41T^{2} \) |
| 43 | \( 1 - 3.76T + 43T^{2} \) |
| 47 | \( 1 - 8.89T + 47T^{2} \) |
| 53 | \( 1 + 4.57T + 53T^{2} \) |
| 59 | \( 1 - 8.08T + 59T^{2} \) |
| 61 | \( 1 - 1.60T + 61T^{2} \) |
| 67 | \( 1 - 11.8T + 67T^{2} \) |
| 71 | \( 1 + 5.18T + 71T^{2} \) |
| 73 | \( 1 - 6.37T + 73T^{2} \) |
| 79 | \( 1 + 1.55T + 79T^{2} \) |
| 83 | \( 1 - 1.82T + 83T^{2} \) |
| 89 | \( 1 - 13.3T + 89T^{2} \) |
| 97 | \( 1 + 6.85T + 97T^{2} \) |
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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
−8.123009287790378440744333669615, −7.61279576212332784977512107121, −6.70151821564282528576747064291, −5.82057665058007877371160771781, −5.23790152598677995628376733937, −4.14805319772455660843227952665, −3.34637829617795853907362233803, −3.12541261135881885072961277903, −2.29532243284194266852861315622, −0.982241393544770158619896432440,
0.982241393544770158619896432440, 2.29532243284194266852861315622, 3.12541261135881885072961277903, 3.34637829617795853907362233803, 4.14805319772455660843227952665, 5.23790152598677995628376733937, 5.82057665058007877371160771781, 6.70151821564282528576747064291, 7.61279576212332784977512107121, 8.123009287790378440744333669615
