L(s) = 1 | + i·2-s − i·3-s − 4-s − 0.942i·5-s + 6-s − i·8-s − 9-s + 0.942·10-s + (1.67 + 2.86i)11-s + i·12-s − 2.81·13-s − 0.942·15-s + 16-s − 4.30·17-s − i·18-s − 2.51·19-s + ⋯ |
L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.421i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + 0.298·10-s + (0.503 + 0.863i)11-s + 0.288i·12-s − 0.781·13-s − 0.243·15-s + 0.250·16-s − 1.04·17-s − 0.235i·18-s − 0.577·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.813 + 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.406672550\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.406672550\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.67 - 2.86i)T \) |
good | 5 | \( 1 + 0.942iT - 5T^{2} \) |
| 13 | \( 1 + 2.81T + 13T^{2} \) |
| 17 | \( 1 + 4.30T + 17T^{2} \) |
| 19 | \( 1 + 2.51T + 19T^{2} \) |
| 23 | \( 1 - 4.33T + 23T^{2} \) |
| 29 | \( 1 + 8.01iT - 29T^{2} \) |
| 31 | \( 1 - 6.05iT - 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 5.20T + 41T^{2} \) |
| 43 | \( 1 + 11.3iT - 43T^{2} \) |
| 47 | \( 1 - 8.96iT - 47T^{2} \) |
| 53 | \( 1 - 4.55T + 53T^{2} \) |
| 59 | \( 1 + 13.4iT - 59T^{2} \) |
| 61 | \( 1 - 2.41T + 61T^{2} \) |
| 67 | \( 1 + 2.22T + 67T^{2} \) |
| 71 | \( 1 + 4.80T + 71T^{2} \) |
| 73 | \( 1 + 4.63T + 73T^{2} \) |
| 79 | \( 1 + 9.26iT - 79T^{2} \) |
| 83 | \( 1 + 8.84T + 83T^{2} \) |
| 89 | \( 1 + 11.5iT - 89T^{2} \) |
| 97 | \( 1 + 12.5iT - 97T^{2} \) |
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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
−8.565562184862029480810800275819, −7.68154980094471585442934106513, −7.05977228645128359892677739376, −6.51439502783390417524441778866, −5.67314821459338556848835173544, −4.64372004116583813659836042185, −4.32568415974254729071700936041, −2.83566276367500264101341994399, −1.85884044551487565249091622406, −0.53580444276612863715454343667,
0.958361605904221417194793974617, 2.45681020922217355147624025071, 3.00149042312010352916054418484, 4.05061027234300528682076389145, 4.63241767504296852696178947607, 5.57050530579704394650476499332, 6.43980003250442222864985223913, 7.20631460936215578076524907283, 8.246243068693301028844878780884, 8.981722137723648800217736078486