L(s) = 1 | + i·2-s + i·3-s − 4-s − 0.516i·5-s − 6-s − i·8-s − 9-s + 0.516·10-s + (−3.09 − 1.19i)11-s − i·12-s − 2.41·13-s + 0.516·15-s + 16-s + 7.29·17-s − i·18-s − 2.57·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s − 0.230i·5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s + 0.163·10-s + (−0.933 − 0.359i)11-s − 0.288i·12-s − 0.669·13-s + 0.133·15-s + 0.250·16-s + 1.77·17-s − 0.235i·18-s − 0.590·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.471872317\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.471872317\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 3 | \( 1 - iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (3.09 + 1.19i)T \) |
good | 5 | \( 1 + 0.516iT - 5T^{2} \) |
| 13 | \( 1 + 2.41T + 13T^{2} \) |
| 17 | \( 1 - 7.29T + 17T^{2} \) |
| 19 | \( 1 + 2.57T + 19T^{2} \) |
| 23 | \( 1 + 1.88T + 23T^{2} \) |
| 29 | \( 1 - 1.33iT - 29T^{2} \) |
| 31 | \( 1 + 4.58iT - 31T^{2} \) |
| 37 | \( 1 - 6.34T + 37T^{2} \) |
| 41 | \( 1 + 2.14T + 41T^{2} \) |
| 43 | \( 1 - 2.68iT - 43T^{2} \) |
| 47 | \( 1 + 1.14iT - 47T^{2} \) |
| 53 | \( 1 - 7.15T + 53T^{2} \) |
| 59 | \( 1 - 1.00iT - 59T^{2} \) |
| 61 | \( 1 + 4.50T + 61T^{2} \) |
| 67 | \( 1 - 8.00T + 67T^{2} \) |
| 71 | \( 1 - 9.05T + 71T^{2} \) |
| 73 | \( 1 - 5.82T + 73T^{2} \) |
| 79 | \( 1 - 17.5iT - 79T^{2} \) |
| 83 | \( 1 + 3.86T + 83T^{2} \) |
| 89 | \( 1 - 14.3iT - 89T^{2} \) |
| 97 | \( 1 - 3.02iT - 97T^{2} \) |
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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
−8.648784440844383233303588805535, −8.041264072388370677197920911150, −7.50007909405480021065845789799, −6.50350788220264297524152964258, −5.60118015640919931145826410625, −5.18533480323507553143333761642, −4.32721356793442951012291739581, −3.40300012218838326823111783952, −2.46142841533454970901069306022, −0.75899766685337359590502888306,
0.68666334002442117833250271917, 1.91688088009325853294622601500, 2.73937926203658370408471162107, 3.48368175124935079051837869445, 4.67722754430155338606890538667, 5.34112113264842670458453784501, 6.18875115562434037839192388260, 7.20053125982164177996702349839, 7.76201663684701920740896462815, 8.422660792865173948396708111702