| L(s) = 1 | − 2-s − 3-s + 4-s − 5-s + 6-s − 8-s + 9-s + 10-s − 12-s − 2·13-s + 15-s + 16-s + 6·17-s − 18-s − 8·19-s − 20-s + 24-s + 25-s + 2·26-s − 27-s + 6·29-s − 30-s + 4·31-s − 32-s − 6·34-s + 36-s − 10·37-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.447·5-s + 0.408·6-s − 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.288·12-s − 0.554·13-s + 0.258·15-s + 1/4·16-s + 1.45·17-s − 0.235·18-s − 1.83·19-s − 0.223·20-s + 0.204·24-s + 1/5·25-s + 0.392·26-s − 0.192·27-s + 1.11·29-s − 0.182·30-s + 0.718·31-s − 0.176·32-s − 1.02·34-s + 1/6·36-s − 1.64·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7641687740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7641687740\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−9.671854035503214752839008315527, −8.526368496609455796791446208261, −8.065715785764846048787694632217, −7.06224867938800248389577305819, −6.45416331724144396581871011345, −5.44446745120546824484392539368, −4.51192569173011558819556455876, −3.39965389205971738905570150540, −2.13174907534489392599600946706, −0.70440313958741481965056319183,
0.70440313958741481965056319183, 2.13174907534489392599600946706, 3.39965389205971738905570150540, 4.51192569173011558819556455876, 5.44446745120546824484392539368, 6.45416331724144396581871011345, 7.06224867938800248389577305819, 8.065715785764846048787694632217, 8.526368496609455796791446208261, 9.671854035503214752839008315527
