| L(s) = 1 | + 2.56·2-s + 3-s + 4.56·4-s + 5-s + 2.56·6-s − 3.12·7-s + 6.56·8-s + 9-s + 2.56·10-s − 5.56·11-s + 4.56·12-s − 2·13-s − 8·14-s + 15-s + 7.68·16-s + 1.12·17-s + 2.56·18-s − 3.12·19-s + 4.56·20-s − 3.12·21-s − 14.2·22-s + 2.43·23-s + 6.56·24-s + 25-s − 5.12·26-s + 27-s − 14.2·28-s + ⋯ |
| L(s) = 1 | + 1.81·2-s + 0.577·3-s + 2.28·4-s + 0.447·5-s + 1.04·6-s − 1.18·7-s + 2.31·8-s + 0.333·9-s + 0.810·10-s − 1.67·11-s + 1.31·12-s − 0.554·13-s − 2.13·14-s + 0.258·15-s + 1.92·16-s + 0.272·17-s + 0.603·18-s − 0.716·19-s + 1.01·20-s − 0.681·21-s − 3.03·22-s + 0.508·23-s + 1.33·24-s + 0.200·25-s − 1.00·26-s + 0.192·27-s − 2.69·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 435 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.113451582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.113451582\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 7 | \( 1 + 3.12T + 7T^{2} \) |
| 11 | \( 1 + 5.56T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 + 3.12T + 19T^{2} \) |
| 23 | \( 1 - 2.43T + 23T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 - 10.6T + 37T^{2} \) |
| 41 | \( 1 - 10.6T + 41T^{2} \) |
| 43 | \( 1 + 4.68T + 43T^{2} \) |
| 47 | \( 1 + 4.87T + 47T^{2} \) |
| 53 | \( 1 + 7.56T + 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 9.12T + 61T^{2} \) |
| 67 | \( 1 - 13.3T + 67T^{2} \) |
| 71 | \( 1 + 11.1T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 + 12T + 79T^{2} \) |
| 83 | \( 1 - 1.56T + 83T^{2} \) |
| 89 | \( 1 + 4.24T + 89T^{2} \) |
| 97 | \( 1 - 6.68T + 97T^{2} \) |
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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
−11.36251397374127041343089626201, −10.34264019006548469411454494862, −9.648021302574650797261891232320, −8.112249447851318817285081961621, −7.08278780101326358501288651886, −6.19888645960561360340900749760, −5.29906685322338603656687617056, −4.28680635450717115076616591496, −2.96591010180264531646197447851, −2.50140308063169562697748210635,
2.50140308063169562697748210635, 2.96591010180264531646197447851, 4.28680635450717115076616591496, 5.29906685322338603656687617056, 6.19888645960561360340900749760, 7.08278780101326358501288651886, 8.112249447851318817285081961621, 9.648021302574650797261891232320, 10.34264019006548469411454494862, 11.36251397374127041343089626201
