L(s) = 1 | + 2-s + 2.25·3-s + 4-s + 5-s + 2.25·6-s + 2.41·7-s + 8-s + 2.10·9-s + 10-s − 4.22·11-s + 2.25·12-s − 13-s + 2.41·14-s + 2.25·15-s + 16-s + 1.47·17-s + 2.10·18-s + 3.33·19-s + 20-s + 5.46·21-s − 4.22·22-s + 1.61·23-s + 2.25·24-s + 25-s − 26-s − 2.02·27-s + 2.41·28-s + ⋯ |
L(s) = 1 | + 0.707·2-s + 1.30·3-s + 0.5·4-s + 0.447·5-s + 0.922·6-s + 0.913·7-s + 0.353·8-s + 0.700·9-s + 0.316·10-s − 1.27·11-s + 0.652·12-s − 0.277·13-s + 0.646·14-s + 0.583·15-s + 0.250·16-s + 0.357·17-s + 0.495·18-s + 0.766·19-s + 0.223·20-s + 1.19·21-s − 0.900·22-s + 0.335·23-s + 0.461·24-s + 0.200·25-s − 0.196·26-s − 0.390·27-s + 0.456·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4030 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(5.700842791\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.700842791\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 31 | \( 1 - T \) |
good | 3 | \( 1 - 2.25T + 3T^{2} \) |
| 7 | \( 1 - 2.41T + 7T^{2} \) |
| 11 | \( 1 + 4.22T + 11T^{2} \) |
| 17 | \( 1 - 1.47T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 - 1.61T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 37 | \( 1 - 4.34T + 37T^{2} \) |
| 41 | \( 1 - 6.70T + 41T^{2} \) |
| 43 | \( 1 - 9.17T + 43T^{2} \) |
| 47 | \( 1 + 4.79T + 47T^{2} \) |
| 53 | \( 1 - 10.0T + 53T^{2} \) |
| 59 | \( 1 - 1.88T + 59T^{2} \) |
| 61 | \( 1 + 14.1T + 61T^{2} \) |
| 67 | \( 1 + 0.593T + 67T^{2} \) |
| 71 | \( 1 - 3.39T + 71T^{2} \) |
| 73 | \( 1 - 0.734T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 6.14T + 83T^{2} \) |
| 89 | \( 1 + 14.2T + 89T^{2} \) |
| 97 | \( 1 + 0.417T + 97T^{2} \) |
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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
−8.197278814717111364212235827309, −7.80309066478163639640782162736, −7.23525390392945741977515668405, −6.06588172807677192607503277968, −5.29624143960044298214666824270, −4.70749287423667088760248570225, −3.75205603291635129486041843642, −2.70778172136651699414502215096, −2.47779221065759808672096932713, −1.28102628392591708266309555309,
1.28102628392591708266309555309, 2.47779221065759808672096932713, 2.70778172136651699414502215096, 3.75205603291635129486041843642, 4.70749287423667088760248570225, 5.29624143960044298214666824270, 6.06588172807677192607503277968, 7.23525390392945741977515668405, 7.80309066478163639640782162736, 8.197278814717111364212235827309