| L(s) = 1 | − 2-s + 0.231·3-s + 4-s + 2.62·5-s − 0.231·6-s + 2.76·7-s − 8-s − 2.94·9-s − 2.62·10-s − 5.25·11-s + 0.231·12-s − 2.76·14-s + 0.606·15-s + 16-s + 4.16·17-s + 2.94·18-s − 19-s + 2.62·20-s + 0.640·21-s + 5.25·22-s + 1.84·23-s − 0.231·24-s + 1.88·25-s − 1.37·27-s + 2.76·28-s − 5.48·29-s − 0.606·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.133·3-s + 0.5·4-s + 1.17·5-s − 0.0944·6-s + 1.04·7-s − 0.353·8-s − 0.982·9-s − 0.829·10-s − 1.58·11-s + 0.0667·12-s − 0.740·14-s + 0.156·15-s + 0.250·16-s + 1.01·17-s + 0.694·18-s − 0.229·19-s + 0.586·20-s + 0.139·21-s + 1.12·22-s + 0.384·23-s − 0.0472·24-s + 0.377·25-s − 0.264·27-s + 0.523·28-s − 1.01·29-s − 0.110·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6422 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.825959458\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.825959458\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 13 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 0.231T + 3T^{2} \) |
| 5 | \( 1 - 2.62T + 5T^{2} \) |
| 7 | \( 1 - 2.76T + 7T^{2} \) |
| 11 | \( 1 + 5.25T + 11T^{2} \) |
| 17 | \( 1 - 4.16T + 17T^{2} \) |
| 23 | \( 1 - 1.84T + 23T^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 31 | \( 1 - 9.97T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 9.70T + 41T^{2} \) |
| 43 | \( 1 - 4.14T + 43T^{2} \) |
| 47 | \( 1 + 2.12T + 47T^{2} \) |
| 53 | \( 1 + 2.38T + 53T^{2} \) |
| 59 | \( 1 - 1.31T + 59T^{2} \) |
| 61 | \( 1 - 11.6T + 61T^{2} \) |
| 67 | \( 1 - 0.0187T + 67T^{2} \) |
| 71 | \( 1 - 2.03T + 71T^{2} \) |
| 73 | \( 1 - 3.46T + 73T^{2} \) |
| 79 | \( 1 - 14.7T + 79T^{2} \) |
| 83 | \( 1 + 9.18T + 83T^{2} \) |
| 89 | \( 1 + 1.80T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.072333123096373328956562536387, −7.67808902159900512237231061688, −6.60787401258768365808947194018, −5.76630386230982030567756559459, −5.40939024071445914932996717973, −4.67552886341280013531980885990, −3.22915130219138296675926449830, −2.49945522694492671555737425803, −1.91189063071213877190237703361, −0.76672131849248544570044114829,
0.76672131849248544570044114829, 1.91189063071213877190237703361, 2.49945522694492671555737425803, 3.22915130219138296675926449830, 4.67552886341280013531980885990, 5.40939024071445914932996717973, 5.76630386230982030567756559459, 6.60787401258768365808947194018, 7.67808902159900512237231061688, 8.072333123096373328956562536387
