| L(s) = 1 | + 1.30·3-s − 4.30·7-s − 1.30·9-s + 11-s + 5·13-s − 3.90·17-s + 19-s − 5.60·21-s + 3.69·23-s − 5.60·27-s − 9.90·29-s + 4.21·31-s + 1.30·33-s + 9.60·37-s + 6.51·39-s + 1.60·41-s + 7.21·43-s + 3·47-s + 11.5·49-s − 5.09·51-s + 2.30·53-s + 1.30·57-s − 0.211·59-s + 2.90·61-s + 5.60·63-s + 4·67-s + 4.81·69-s + ⋯ |
| L(s) = 1 | + 0.752·3-s − 1.62·7-s − 0.434·9-s + 0.301·11-s + 1.38·13-s − 0.947·17-s + 0.229·19-s − 1.22·21-s + 0.770·23-s − 1.07·27-s − 1.83·29-s + 0.756·31-s + 0.226·33-s + 1.57·37-s + 1.04·39-s + 0.250·41-s + 1.09·43-s + 0.437·47-s + 1.64·49-s − 0.712·51-s + 0.316·53-s + 0.172·57-s − 0.0274·59-s + 0.372·61-s + 0.706·63-s + 0.488·67-s + 0.579·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.872673204\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.872673204\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - 1.30T + 3T^{2} \) |
| 7 | \( 1 + 4.30T + 7T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 + 3.90T + 17T^{2} \) |
| 19 | \( 1 - T + 19T^{2} \) |
| 23 | \( 1 - 3.69T + 23T^{2} \) |
| 29 | \( 1 + 9.90T + 29T^{2} \) |
| 31 | \( 1 - 4.21T + 31T^{2} \) |
| 37 | \( 1 - 9.60T + 37T^{2} \) |
| 41 | \( 1 - 1.60T + 41T^{2} \) |
| 43 | \( 1 - 7.21T + 43T^{2} \) |
| 47 | \( 1 - 3T + 47T^{2} \) |
| 53 | \( 1 - 2.30T + 53T^{2} \) |
| 59 | \( 1 + 0.211T + 59T^{2} \) |
| 61 | \( 1 - 2.90T + 61T^{2} \) |
| 67 | \( 1 - 4T + 67T^{2} \) |
| 71 | \( 1 + 4.60T + 71T^{2} \) |
| 73 | \( 1 - 2.90T + 73T^{2} \) |
| 79 | \( 1 - 0.0916T + 79T^{2} \) |
| 83 | \( 1 + 14.5T + 83T^{2} \) |
| 89 | \( 1 - 5.30T + 89T^{2} \) |
| 97 | \( 1 - 11.6T + 97T^{2} \) |
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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
−8.563555901948273760083052683011, −7.64740536756075022827535011153, −6.90156588772389336373932056727, −6.09843854247256797290281611446, −5.75774114840410178883971131128, −4.31618301600989089615911244990, −3.61354332413086338769244951268, −3.02842212827073324960033697239, −2.18716396967952105233919936447, −0.72030134262674648043428251437,
0.72030134262674648043428251437, 2.18716396967952105233919936447, 3.02842212827073324960033697239, 3.61354332413086338769244951268, 4.31618301600989089615911244990, 5.75774114840410178883971131128, 6.09843854247256797290281611446, 6.90156588772389336373932056727, 7.64740536756075022827535011153, 8.563555901948273760083052683011
