| L(s) = 1 | + 2.11·2-s − 3.18·3-s + 2.49·4-s + 1.61·5-s − 6.74·6-s + 3.36·7-s + 1.03·8-s + 7.11·9-s + 3.42·10-s + 11-s − 7.92·12-s + 0.557·13-s + 7.12·14-s − 5.13·15-s − 2.77·16-s − 3.12·17-s + 15.0·18-s − 1.99·19-s + 4.02·20-s − 10.6·21-s + 2.11·22-s + 9.11·23-s − 3.30·24-s − 2.39·25-s + 1.18·26-s − 13.1·27-s + 8.37·28-s + ⋯ |
| L(s) = 1 | + 1.49·2-s − 1.83·3-s + 1.24·4-s + 0.722·5-s − 2.75·6-s + 1.27·7-s + 0.367·8-s + 2.37·9-s + 1.08·10-s + 0.301·11-s − 2.28·12-s + 0.154·13-s + 1.90·14-s − 1.32·15-s − 0.694·16-s − 0.757·17-s + 3.55·18-s − 0.457·19-s + 0.899·20-s − 2.33·21-s + 0.451·22-s + 1.90·23-s − 0.674·24-s − 0.478·25-s + 0.231·26-s − 2.52·27-s + 1.58·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.798795968\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.798795968\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 2.11T + 2T^{2} \) |
| 3 | \( 1 + 3.18T + 3T^{2} \) |
| 5 | \( 1 - 1.61T + 5T^{2} \) |
| 7 | \( 1 - 3.36T + 7T^{2} \) |
| 13 | \( 1 - 0.557T + 13T^{2} \) |
| 17 | \( 1 + 3.12T + 17T^{2} \) |
| 19 | \( 1 + 1.99T + 19T^{2} \) |
| 23 | \( 1 - 9.11T + 23T^{2} \) |
| 29 | \( 1 - 2.14T + 29T^{2} \) |
| 31 | \( 1 - 7.14T + 31T^{2} \) |
| 37 | \( 1 - 4.22T + 37T^{2} \) |
| 41 | \( 1 - 4.84T + 41T^{2} \) |
| 43 | \( 1 - 2.12T + 43T^{2} \) |
| 47 | \( 1 - 2.67T + 47T^{2} \) |
| 53 | \( 1 + 2.57T + 53T^{2} \) |
| 59 | \( 1 + 7.07T + 59T^{2} \) |
| 61 | \( 1 - 8.43T + 61T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 + 8.23T + 73T^{2} \) |
| 79 | \( 1 - 5.87T + 79T^{2} \) |
| 83 | \( 1 + 5.87T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 6.22T + 97T^{2} \) |
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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
−9.850695003968239577531280025455, −8.775515648574968951049125613940, −7.40885809980525503367062775977, −6.48547136375272626679256661760, −6.10384111682753211897796683203, −5.15955173394583661504603438144, −4.80360319551024063046686652647, −4.07798715773448782862557019707, −2.38841640829164104578193062044, −1.15294663862356821733989532147,
1.15294663862356821733989532147, 2.38841640829164104578193062044, 4.07798715773448782862557019707, 4.80360319551024063046686652647, 5.15955173394583661504603438144, 6.10384111682753211897796683203, 6.48547136375272626679256661760, 7.40885809980525503367062775977, 8.775515648574968951049125613940, 9.850695003968239577531280025455
