| L(s) = 1 | + 1.73·2-s − 2.96·3-s + 1.00·4-s − 4.01·5-s − 5.13·6-s − 3.85·7-s − 1.72·8-s + 5.78·9-s − 6.96·10-s − 6.28·11-s − 2.97·12-s + 0.685·13-s − 6.68·14-s + 11.9·15-s − 4.99·16-s + 3.45·17-s + 10.0·18-s − 5.76·19-s − 4.03·20-s + 11.4·21-s − 10.8·22-s − 2.39·23-s + 5.11·24-s + 11.1·25-s + 1.18·26-s − 8.24·27-s − 3.87·28-s + ⋯ |
| L(s) = 1 | + 1.22·2-s − 1.71·3-s + 0.501·4-s − 1.79·5-s − 2.09·6-s − 1.45·7-s − 0.610·8-s + 1.92·9-s − 2.20·10-s − 1.89·11-s − 0.858·12-s + 0.190·13-s − 1.78·14-s + 3.07·15-s − 1.24·16-s + 0.838·17-s + 2.36·18-s − 1.32·19-s − 0.901·20-s + 2.49·21-s − 2.32·22-s − 0.499·23-s + 1.04·24-s + 2.22·25-s + 0.233·26-s − 1.58·27-s − 0.731·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1511 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.005527452170\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.005527452170\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1511 | \( 1 - T \) |
| good | 2 | \( 1 - 1.73T + 2T^{2} \) |
| 3 | \( 1 + 2.96T + 3T^{2} \) |
| 5 | \( 1 + 4.01T + 5T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 + 6.28T + 11T^{2} \) |
| 13 | \( 1 - 0.685T + 13T^{2} \) |
| 17 | \( 1 - 3.45T + 17T^{2} \) |
| 19 | \( 1 + 5.76T + 19T^{2} \) |
| 23 | \( 1 + 2.39T + 23T^{2} \) |
| 29 | \( 1 + 5.85T + 29T^{2} \) |
| 31 | \( 1 - 0.445T + 31T^{2} \) |
| 37 | \( 1 + 1.49T + 37T^{2} \) |
| 41 | \( 1 + 5.58T + 41T^{2} \) |
| 43 | \( 1 - 2.69T + 43T^{2} \) |
| 47 | \( 1 + 7.40T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 + 14.5T + 59T^{2} \) |
| 61 | \( 1 + 1.28T + 61T^{2} \) |
| 67 | \( 1 + 6.79T + 67T^{2} \) |
| 71 | \( 1 - 9.87T + 71T^{2} \) |
| 73 | \( 1 + 9.43T + 73T^{2} \) |
| 79 | \( 1 + 3.24T + 79T^{2} \) |
| 83 | \( 1 + 9.09T + 83T^{2} \) |
| 89 | \( 1 - 3.77T + 89T^{2} \) |
| 97 | \( 1 + 9.81T + 97T^{2} \) |
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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
−9.842686725777299985071486283728, −8.467799853085374035566010398525, −7.50520295841538032184400140491, −6.76650746092313446557602228426, −5.95575227949030210902301972020, −5.32253077545093587400776362544, −4.48379851148798081873957814375, −3.76963996131050173284038326790, −2.92960465202715011133224424625, −0.04400737424629402157294873044,
0.04400737424629402157294873044, 2.92960465202715011133224424625, 3.76963996131050173284038326790, 4.48379851148798081873957814375, 5.32253077545093587400776362544, 5.95575227949030210902301972020, 6.76650746092313446557602228426, 7.50520295841538032184400140491, 8.467799853085374035566010398525, 9.842686725777299985071486283728
