| L(s) = 1 | + i·2-s + 3-s − 4-s − 3i·5-s + i·6-s + i·7-s − i·8-s − 2·9-s + 3·10-s − 12-s − 14-s − 3i·15-s + 16-s − 6·17-s − 2i·18-s + 4i·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577·3-s − 0.5·4-s − 1.34i·5-s + 0.408i·6-s + 0.377i·7-s − 0.353i·8-s − 0.666·9-s + 0.948·10-s − 0.288·12-s − 0.267·14-s − 0.774i·15-s + 0.250·16-s − 1.45·17-s − 0.471i·18-s + 0.917i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2366 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8223288376\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8223288376\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 7 | \( 1 - iT \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - T + 3T^{2} \) |
| 5 | \( 1 + 3iT - 5T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 17 | \( 1 + 6T + 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 3T + 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 10iT - 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 8T + 43T^{2} \) |
| 47 | \( 1 - 6iT - 47T^{2} \) |
| 53 | \( 1 - 12T + 53T^{2} \) |
| 59 | \( 1 - 3iT - 59T^{2} \) |
| 61 | \( 1 - 11T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 3iT - 71T^{2} \) |
| 73 | \( 1 - 2iT - 73T^{2} \) |
| 79 | \( 1 + 4T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 + 2iT - 97T^{2} \) |
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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
−8.909915704917064402687804257265, −8.399608969780897245062336445500, −8.236233912801962550110668522713, −6.95035917319938559553346881557, −6.17206142963978360485214763723, −5.29770523788072810842003443263, −4.70274768545166679904637437757, −3.77788662160686800284280997790, −2.60908862218914052975663119214, −1.36379926784829668720155680597,
0.25137329983303818760358120418, 2.26440903836618797729945488190, 2.56434308846424298556299870895, 3.60669837191385399229260101203, 4.27545031017248526225183050146, 5.49205143778407696733814779295, 6.51057513577213765728360293204, 7.08592560916749619839675265862, 8.056315283779918121024950844526, 8.722427542424209511413133402114
