| L(s) = 1 | − 0.858·2-s − 3-s − 1.26·4-s − 4.02·5-s + 0.858·6-s + 1.69·7-s + 2.80·8-s + 9-s + 3.45·10-s + 4.28·11-s + 1.26·12-s + 13-s − 1.45·14-s + 4.02·15-s + 0.120·16-s + 7.37·17-s − 0.858·18-s + 0.247·19-s + 5.08·20-s − 1.69·21-s − 3.68·22-s + 7.02·23-s − 2.80·24-s + 11.2·25-s − 0.858·26-s − 27-s − 2.13·28-s + ⋯ |
| L(s) = 1 | − 0.607·2-s − 0.577·3-s − 0.631·4-s − 1.80·5-s + 0.350·6-s + 0.639·7-s + 0.990·8-s + 0.333·9-s + 1.09·10-s + 1.29·11-s + 0.364·12-s + 0.277·13-s − 0.387·14-s + 1.03·15-s + 0.0301·16-s + 1.78·17-s − 0.202·18-s + 0.0568·19-s + 1.13·20-s − 0.368·21-s − 0.785·22-s + 1.46·23-s − 0.571·24-s + 2.24·25-s − 0.168·26-s − 0.192·27-s − 0.403·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8887908627\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8887908627\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 + T \) |
| good | 2 | \( 1 + 0.858T + 2T^{2} \) |
| 5 | \( 1 + 4.02T + 5T^{2} \) |
| 7 | \( 1 - 1.69T + 7T^{2} \) |
| 11 | \( 1 - 4.28T + 11T^{2} \) |
| 17 | \( 1 - 7.37T + 17T^{2} \) |
| 19 | \( 1 - 0.247T + 19T^{2} \) |
| 23 | \( 1 - 7.02T + 23T^{2} \) |
| 29 | \( 1 + 2.37T + 29T^{2} \) |
| 31 | \( 1 - 7.06T + 31T^{2} \) |
| 37 | \( 1 + 0.651T + 37T^{2} \) |
| 41 | \( 1 - 4.11T + 41T^{2} \) |
| 43 | \( 1 + 4.16T + 43T^{2} \) |
| 47 | \( 1 - 2.62T + 47T^{2} \) |
| 53 | \( 1 + 7.05T + 53T^{2} \) |
| 59 | \( 1 + 10.9T + 59T^{2} \) |
| 61 | \( 1 - 9.22T + 61T^{2} \) |
| 67 | \( 1 - 0.887T + 67T^{2} \) |
| 71 | \( 1 - 12.7T + 71T^{2} \) |
| 73 | \( 1 - 8.42T + 73T^{2} \) |
| 79 | \( 1 - 5.27T + 79T^{2} \) |
| 83 | \( 1 - 11.2T + 83T^{2} \) |
| 89 | \( 1 - 0.789T + 89T^{2} \) |
| 97 | \( 1 + 14.1T + 97T^{2} \) |
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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
−8.248455225199817323973293848312, −7.903409464791945592577068117568, −7.23349175646669621160856500485, −6.43982565552016961398640764502, −5.18282410656816844010444562347, −4.69466898526352522961025726595, −3.86926309163997259327109081590, −3.33427687063631898103581171065, −1.27724393392296465274660646952, −0.75428285198665737373365684782,
0.75428285198665737373365684782, 1.27724393392296465274660646952, 3.33427687063631898103581171065, 3.86926309163997259327109081590, 4.69466898526352522961025726595, 5.18282410656816844010444562347, 6.43982565552016961398640764502, 7.23349175646669621160856500485, 7.903409464791945592577068117568, 8.248455225199817323973293848312
