| L(s) = 1 | + 2.13·2-s − 2.20·3-s + 2.56·4-s − 1.38·5-s − 4.71·6-s + 5.12·7-s + 1.19·8-s + 1.88·9-s − 2.95·10-s + 11-s − 5.65·12-s + 10.9·14-s + 3.05·15-s − 2.56·16-s − 0.645·17-s + 4.02·18-s + 5.71·19-s − 3.54·20-s − 11.3·21-s + 2.13·22-s − 2.40·23-s − 2.64·24-s − 3.08·25-s + 2.46·27-s + 13.1·28-s + 2.88·29-s + 6.53·30-s + ⋯ |
| L(s) = 1 | + 1.51·2-s − 1.27·3-s + 1.28·4-s − 0.619·5-s − 1.92·6-s + 1.93·7-s + 0.423·8-s + 0.627·9-s − 0.935·10-s + 0.301·11-s − 1.63·12-s + 2.92·14-s + 0.789·15-s − 0.640·16-s − 0.156·17-s + 0.947·18-s + 1.31·19-s − 0.792·20-s − 2.46·21-s + 0.455·22-s − 0.502·23-s − 0.540·24-s − 0.616·25-s + 0.475·27-s + 2.47·28-s + 0.535·29-s + 1.19·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.858417238\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.858417238\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 2.13T + 2T^{2} \) |
| 3 | \( 1 + 2.20T + 3T^{2} \) |
| 5 | \( 1 + 1.38T + 5T^{2} \) |
| 7 | \( 1 - 5.12T + 7T^{2} \) |
| 17 | \( 1 + 0.645T + 17T^{2} \) |
| 19 | \( 1 - 5.71T + 19T^{2} \) |
| 23 | \( 1 + 2.40T + 23T^{2} \) |
| 29 | \( 1 - 2.88T + 29T^{2} \) |
| 31 | \( 1 - 3.43T + 31T^{2} \) |
| 37 | \( 1 - 10.2T + 37T^{2} \) |
| 41 | \( 1 + 3.99T + 41T^{2} \) |
| 43 | \( 1 - 7.08T + 43T^{2} \) |
| 47 | \( 1 - 11.7T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 + 0.0415T + 59T^{2} \) |
| 61 | \( 1 + 4.29T + 61T^{2} \) |
| 67 | \( 1 - 14.4T + 67T^{2} \) |
| 71 | \( 1 - 4.10T + 71T^{2} \) |
| 73 | \( 1 - 4.85T + 73T^{2} \) |
| 79 | \( 1 - 3.30T + 79T^{2} \) |
| 83 | \( 1 - 3.87T + 83T^{2} \) |
| 89 | \( 1 - 5.70T + 89T^{2} \) |
| 97 | \( 1 + 0.622T + 97T^{2} \) |
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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
−9.257655568233209303134402890571, −8.077696835388999741381660392469, −7.53556280079850727748970252350, −6.46959791428397762718342950617, −5.73480308809142319978637079418, −5.06379690482168320467928826013, −4.54806076760642776096793415344, −3.83287943339158845369428986348, −2.43853530358166760607933408190, −1.03601625879472138094505889658,
1.03601625879472138094505889658, 2.43853530358166760607933408190, 3.83287943339158845369428986348, 4.54806076760642776096793415344, 5.06379690482168320467928826013, 5.73480308809142319978637079418, 6.46959791428397762718342950617, 7.53556280079850727748970252350, 8.077696835388999741381660392469, 9.257655568233209303134402890571
