| L(s) = 1 | − 2.13·2-s + 2.55·4-s − 5-s + 4.82·7-s − 1.17·8-s + 2.13·10-s + 4.26·13-s − 10.2·14-s − 2.59·16-s − 4.64·17-s − 6.37·19-s − 2.55·20-s + 5.14·23-s + 25-s − 9.10·26-s + 12.3·28-s − 4.26·29-s − 6.39·31-s + 7.88·32-s + 9.90·34-s − 4.82·35-s + 6.14·37-s + 13.5·38-s + 1.17·40-s − 3.46·41-s − 1.55·43-s − 10.9·46-s + ⋯ |
| L(s) = 1 | − 1.50·2-s + 1.27·4-s − 0.447·5-s + 1.82·7-s − 0.416·8-s + 0.674·10-s + 1.18·13-s − 2.74·14-s − 0.647·16-s − 1.12·17-s − 1.46·19-s − 0.570·20-s + 1.07·23-s + 0.200·25-s − 1.78·26-s + 2.32·28-s − 0.792·29-s − 1.14·31-s + 1.39·32-s + 1.69·34-s − 0.814·35-s + 1.00·37-s + 2.20·38-s + 0.186·40-s − 0.541·41-s − 0.236·43-s − 1.61·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9693632407\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9693632407\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.13T + 2T^{2} \) |
| 7 | \( 1 - 4.82T + 7T^{2} \) |
| 13 | \( 1 - 4.26T + 13T^{2} \) |
| 17 | \( 1 + 4.64T + 17T^{2} \) |
| 19 | \( 1 + 6.37T + 19T^{2} \) |
| 23 | \( 1 - 5.14T + 23T^{2} \) |
| 29 | \( 1 + 4.26T + 29T^{2} \) |
| 31 | \( 1 + 6.39T + 31T^{2} \) |
| 37 | \( 1 - 6.14T + 37T^{2} \) |
| 41 | \( 1 + 3.46T + 41T^{2} \) |
| 43 | \( 1 + 1.55T + 43T^{2} \) |
| 47 | \( 1 + 5.35T + 47T^{2} \) |
| 53 | \( 1 + 2.24T + 53T^{2} \) |
| 59 | \( 1 - 13.5T + 59T^{2} \) |
| 61 | \( 1 + 6.80T + 61T^{2} \) |
| 67 | \( 1 - 6.14T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 5.62T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 + 6.17T + 83T^{2} \) |
| 89 | \( 1 - 1.39T + 89T^{2} \) |
| 97 | \( 1 - 3.93T + 97T^{2} \) |
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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
−8.198781617374548274098248242570, −7.84320367977548988507413102104, −6.99162298372315706374045198536, −6.37398923478326193254115851933, −5.20121029235591270857299876230, −4.50865422147536420017683192125, −3.75086133741116756570626327906, −2.22526224566019542574758983700, −1.70567574809066292596865643005, −0.68989211239839315234940367968,
0.68989211239839315234940367968, 1.70567574809066292596865643005, 2.22526224566019542574758983700, 3.75086133741116756570626327906, 4.50865422147536420017683192125, 5.20121029235591270857299876230, 6.37398923478326193254115851933, 6.99162298372315706374045198536, 7.84320367977548988507413102104, 8.198781617374548274098248242570
