| L(s) = 1 | + 2.22·2-s − 1.03·3-s + 2.94·4-s + 5-s − 2.30·6-s − 2.02·7-s + 2.09·8-s − 1.92·9-s + 2.22·10-s + 0.0848·11-s − 3.04·12-s − 5.72·13-s − 4.50·14-s − 1.03·15-s − 1.23·16-s − 2.53·17-s − 4.27·18-s + 2.94·20-s + 2.10·21-s + 0.188·22-s − 0.309·23-s − 2.16·24-s + 25-s − 12.7·26-s + 5.10·27-s − 5.95·28-s − 2.62·29-s + ⋯ |
| L(s) = 1 | + 1.57·2-s − 0.598·3-s + 1.47·4-s + 0.447·5-s − 0.941·6-s − 0.765·7-s + 0.739·8-s − 0.641·9-s + 0.702·10-s + 0.0255·11-s − 0.880·12-s − 1.58·13-s − 1.20·14-s − 0.267·15-s − 0.308·16-s − 0.613·17-s − 1.00·18-s + 0.657·20-s + 0.458·21-s + 0.0401·22-s − 0.0644·23-s − 0.442·24-s + 0.200·25-s − 2.49·26-s + 0.982·27-s − 1.12·28-s − 0.487·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1805 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 2 | \( 1 - 2.22T + 2T^{2} \) |
| 3 | \( 1 + 1.03T + 3T^{2} \) |
| 7 | \( 1 + 2.02T + 7T^{2} \) |
| 11 | \( 1 - 0.0848T + 11T^{2} \) |
| 13 | \( 1 + 5.72T + 13T^{2} \) |
| 17 | \( 1 + 2.53T + 17T^{2} \) |
| 23 | \( 1 + 0.309T + 23T^{2} \) |
| 29 | \( 1 + 2.62T + 29T^{2} \) |
| 31 | \( 1 + 8.07T + 31T^{2} \) |
| 37 | \( 1 - 5.01T + 37T^{2} \) |
| 41 | \( 1 - 5.88T + 41T^{2} \) |
| 43 | \( 1 - 0.650T + 43T^{2} \) |
| 47 | \( 1 - 6.90T + 47T^{2} \) |
| 53 | \( 1 + 14.5T + 53T^{2} \) |
| 59 | \( 1 + 7.47T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 + 8.88T + 67T^{2} \) |
| 71 | \( 1 - 14.3T + 71T^{2} \) |
| 73 | \( 1 - 10.8T + 73T^{2} \) |
| 79 | \( 1 + 0.115T + 79T^{2} \) |
| 83 | \( 1 + 2.97T + 83T^{2} \) |
| 89 | \( 1 - 11.1T + 89T^{2} \) |
| 97 | \( 1 - 0.225T + 97T^{2} \) |
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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
−9.175066753268085903364295117289, −7.74962010138005213403461139909, −6.81374334565036585024875216629, −6.25400904025223471773055385752, −5.48349497515999516884079188621, −4.95455192902197370668103951074, −3.98968675267109176659970492658, −2.93365560858664623332066254122, −2.24002163454168145107926660245, 0,
2.24002163454168145107926660245, 2.93365560858664623332066254122, 3.98968675267109176659970492658, 4.95455192902197370668103951074, 5.48349497515999516884079188621, 6.25400904025223471773055385752, 6.81374334565036585024875216629, 7.74962010138005213403461139909, 9.175066753268085903364295117289
