| L(s) = 1 | − 2.29·2-s − 3-s + 3.25·4-s − 0.692·5-s + 2.29·6-s − 7-s − 2.88·8-s + 9-s + 1.58·10-s + 6.01·11-s − 3.25·12-s + 2.29·14-s + 0.692·15-s + 0.103·16-s − 7.99·17-s − 2.29·18-s + 3.00·19-s − 2.25·20-s + 21-s − 13.7·22-s + 0.653·23-s + 2.88·24-s − 4.52·25-s − 27-s − 3.25·28-s + 9.28·29-s − 1.58·30-s + ⋯ |
| L(s) = 1 | − 1.62·2-s − 0.577·3-s + 1.62·4-s − 0.309·5-s + 0.936·6-s − 0.377·7-s − 1.02·8-s + 0.333·9-s + 0.502·10-s + 1.81·11-s − 0.940·12-s + 0.612·14-s + 0.178·15-s + 0.0259·16-s − 1.93·17-s − 0.540·18-s + 0.690·19-s − 0.504·20-s + 0.218·21-s − 2.94·22-s + 0.136·23-s + 0.589·24-s − 0.904·25-s − 0.192·27-s − 0.615·28-s + 1.72·29-s − 0.289·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3549 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5674915811\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5674915811\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 + 2.29T + 2T^{2} \) |
| 5 | \( 1 + 0.692T + 5T^{2} \) |
| 11 | \( 1 - 6.01T + 11T^{2} \) |
| 17 | \( 1 + 7.99T + 17T^{2} \) |
| 19 | \( 1 - 3.00T + 19T^{2} \) |
| 23 | \( 1 - 0.653T + 23T^{2} \) |
| 29 | \( 1 - 9.28T + 29T^{2} \) |
| 31 | \( 1 - 1.47T + 31T^{2} \) |
| 37 | \( 1 - 4.03T + 37T^{2} \) |
| 41 | \( 1 - 9.18T + 41T^{2} \) |
| 43 | \( 1 + 8.12T + 43T^{2} \) |
| 47 | \( 1 + 0.664T + 47T^{2} \) |
| 53 | \( 1 - 10.3T + 53T^{2} \) |
| 59 | \( 1 + 2.04T + 59T^{2} \) |
| 61 | \( 1 - 1.38T + 61T^{2} \) |
| 67 | \( 1 + 7.38T + 67T^{2} \) |
| 71 | \( 1 + 1.00T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 6.08T + 79T^{2} \) |
| 83 | \( 1 + 0.377T + 83T^{2} \) |
| 89 | \( 1 - 6.63T + 89T^{2} \) |
| 97 | \( 1 - 0.346T + 97T^{2} \) |
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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
−8.807714805578183308321624950265, −7.949191087602743389898758020817, −7.05799294879622174396902393861, −6.61039798926800480833917348633, −6.05951095041182068090730442653, −4.61438402033109445070518781891, −3.98827114893013365931867990692, −2.64245671947669201464684920670, −1.53218616451612193217832450466, −0.61730079801775600997960199717,
0.61730079801775600997960199717, 1.53218616451612193217832450466, 2.64245671947669201464684920670, 3.98827114893013365931867990692, 4.61438402033109445070518781891, 6.05951095041182068090730442653, 6.61039798926800480833917348633, 7.05799294879622174396902393861, 7.949191087602743389898758020817, 8.807714805578183308321624950265
