| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + i·8-s + 2·9-s + 3·11-s + i·12-s − 2i·13-s + 16-s + 3i·17-s − 2i·18-s − 7·19-s − 3i·22-s + 24-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.353i·8-s + 0.666·9-s + 0.904·11-s + 0.288i·12-s − 0.554i·13-s + 0.250·16-s + 0.727i·17-s − 0.471i·18-s − 1.60·19-s − 0.639i·22-s + 0.204·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.816632797\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.816632797\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + 2iT - 13T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 + 7T + 19T^{2} \) |
| 23 | \( 1 - 23T^{2} \) |
| 29 | \( 1 - 6T + 29T^{2} \) |
| 31 | \( 1 - 4T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 9T + 41T^{2} \) |
| 43 | \( 1 - 8iT - 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 12iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 10T + 61T^{2} \) |
| 67 | \( 1 - 7iT - 67T^{2} \) |
| 71 | \( 1 - 6T + 71T^{2} \) |
| 73 | \( 1 + 5iT - 73T^{2} \) |
| 79 | \( 1 + 14T + 79T^{2} \) |
| 83 | \( 1 - 9iT - 83T^{2} \) |
| 89 | \( 1 + 15T + 89T^{2} \) |
| 97 | \( 1 + 10iT - 97T^{2} \) |
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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
−8.490532124872863079464037018826, −8.251577106685596022461795390188, −7.04842833833590880277776721134, −6.48129634868578868636556016924, −5.60306702001252148003430190571, −4.33478590734276780168099009521, −3.95806924599184681134781725135, −2.63760473033918254025977603676, −1.76616264277131175403557650458, −0.73743426868524249296873055799,
1.13931559641777263878283601826, 2.58014810226331224510076452612, 3.98041734576428427686310747813, 4.34178168622102925364911426728, 5.16460923396670724988183339617, 6.35922549668731555721178004746, 6.70162105190977187375468416688, 7.58151874902949358064537688703, 8.566767828534576884876706659597, 9.060403820093175901978861093502
