| L(s) = 1 | + 0.0526·2-s − 1.28·3-s − 1.99·4-s − 0.451·5-s − 0.0674·6-s − 0.210·8-s − 1.35·9-s − 0.0237·10-s − 4.14·11-s + 2.56·12-s + 0.988·13-s + 0.579·15-s + 3.98·16-s + 17-s − 0.0713·18-s + 2.62·19-s + 0.902·20-s − 0.218·22-s − 1.66·23-s + 0.269·24-s − 4.79·25-s + 0.0520·26-s + 5.58·27-s + 3.20·29-s + 0.0304·30-s + 8.31·31-s + 0.630·32-s + ⋯ |
| L(s) = 1 | + 0.0372·2-s − 0.740·3-s − 0.998·4-s − 0.202·5-s − 0.0275·6-s − 0.0743·8-s − 0.452·9-s − 0.00751·10-s − 1.25·11-s + 0.739·12-s + 0.274·13-s + 0.149·15-s + 0.995·16-s + 0.242·17-s − 0.0168·18-s + 0.601·19-s + 0.201·20-s − 0.0465·22-s − 0.348·23-s + 0.0550·24-s − 0.959·25-s + 0.0102·26-s + 1.07·27-s + 0.594·29-s + 0.00556·30-s + 1.49·31-s + 0.111·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 833 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6587609922\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6587609922\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 2 | \( 1 - 0.0526T + 2T^{2} \) |
| 3 | \( 1 + 1.28T + 3T^{2} \) |
| 5 | \( 1 + 0.451T + 5T^{2} \) |
| 11 | \( 1 + 4.14T + 11T^{2} \) |
| 13 | \( 1 - 0.988T + 13T^{2} \) |
| 19 | \( 1 - 2.62T + 19T^{2} \) |
| 23 | \( 1 + 1.66T + 23T^{2} \) |
| 29 | \( 1 - 3.20T + 29T^{2} \) |
| 31 | \( 1 - 8.31T + 31T^{2} \) |
| 37 | \( 1 - 5.35T + 37T^{2} \) |
| 41 | \( 1 + 3.74T + 41T^{2} \) |
| 43 | \( 1 - 0.271T + 43T^{2} \) |
| 47 | \( 1 - 1.87T + 47T^{2} \) |
| 53 | \( 1 + 4.61T + 53T^{2} \) |
| 59 | \( 1 - 11.1T + 59T^{2} \) |
| 61 | \( 1 - 6.31T + 61T^{2} \) |
| 67 | \( 1 - 11.0T + 67T^{2} \) |
| 71 | \( 1 - 10.4T + 71T^{2} \) |
| 73 | \( 1 - 6.74T + 73T^{2} \) |
| 79 | \( 1 + 0.0703T + 79T^{2} \) |
| 83 | \( 1 - 3.62T + 83T^{2} \) |
| 89 | \( 1 - 6.43T + 89T^{2} \) |
| 97 | \( 1 + 3.59T + 97T^{2} \) |
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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
−10.16245263373035394372131671865, −9.548132013501225224110805797703, −8.229049381498166724393487632303, −8.043116137478009315499696580311, −6.57556752136677068115469870398, −5.55657031141488642634350877776, −5.07285683341805482184152037577, −3.97210365532531782649204173589, −2.76020340630797526681409313403, −0.66931407315206132147417338182,
0.66931407315206132147417338182, 2.76020340630797526681409313403, 3.97210365532531782649204173589, 5.07285683341805482184152037577, 5.55657031141488642634350877776, 6.57556752136677068115469870398, 8.043116137478009315499696580311, 8.229049381498166724393487632303, 9.548132013501225224110805797703, 10.16245263373035394372131671865
