| L(s) = 1 | + (−1.41 + i)3-s + 1.41i·5-s − i·7-s + (1.00 − 2.82i)9-s + 1.41·11-s + 6·13-s + (−1.41 − 2.00i)15-s + 1.41i·17-s + 4i·19-s + (1 + 1.41i)21-s − 7.07·23-s + 2.99·25-s + (1.41 + 5.00i)27-s + 8.48i·29-s + 2i·31-s + ⋯ |
| L(s) = 1 | + (−0.816 + 0.577i)3-s + 0.632i·5-s − 0.377i·7-s + (0.333 − 0.942i)9-s + 0.426·11-s + 1.66·13-s + (−0.365 − 0.516i)15-s + 0.342i·17-s + 0.917i·19-s + (0.218 + 0.308i)21-s − 1.47·23-s + 0.599·25-s + (0.272 + 0.962i)27-s + 1.57i·29-s + 0.359i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 672 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.877675 + 0.739914i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.877675 + 0.739914i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.41 - i)T \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 - 1.41iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 7.07T + 23T^{2} \) |
| 29 | \( 1 - 8.48iT - 29T^{2} \) |
| 31 | \( 1 - 2iT - 31T^{2} \) |
| 37 | \( 1 + 4T + 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 16iT - 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 - 14T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 - 8.48T + 83T^{2} \) |
| 89 | \( 1 - 7.07iT - 89T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−10.60669516030233059335209844446, −10.20639985335889591031378794870, −9.056532531878420881218316939245, −8.167212773613111909765310596915, −6.84176252730388801822811237722, −6.28178835773177346930465721014, −5.36982684689779848746050659130, −4.01493036803378932483330905584, −3.46180807729881421563979511984, −1.37957040085384193193361657961,
0.78275659100519606258338743859, 2.12091020995443491702709848618, 3.88572930058800319002859268860, 4.94589299982810611379148849924, 5.95323126076860277069743914398, 6.50573491882623869180484053423, 7.72142211051777158833357119158, 8.539786344947961811886504662070, 9.355906069603127123067578867945, 10.51098387511703739317886139374
