L(s) = 1 | + i·2-s + (−0.707 + 0.707i)3-s − 4-s + (−1.41 + 1.41i)5-s + (−0.707 − 0.707i)6-s − i·8-s − 1.00i·9-s + (−1.41 − 1.41i)10-s + (2.82 + 2.82i)11-s + (0.707 − 0.707i)12-s + 2·13-s − 2.00i·15-s + 16-s + 1.00·18-s + 4i·19-s + (1.41 − 1.41i)20-s + ⋯ |
L(s) = 1 | + 0.707i·2-s + (−0.408 + 0.408i)3-s − 0.5·4-s + (−0.632 + 0.632i)5-s + (−0.288 − 0.288i)6-s − 0.353i·8-s − 0.333i·9-s + (−0.447 − 0.447i)10-s + (0.852 + 0.852i)11-s + (0.204 − 0.204i)12-s + 0.554·13-s − 0.516i·15-s + 0.250·16-s + 0.235·18-s + 0.917i·19-s + (0.316 − 0.316i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.805 + 0.591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1734 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.805 + 0.591i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7768861598\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7768861598\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.707 - 0.707i)T \) |
| 17 | \( 1 \) |
good | 5 | \( 1 + (1.41 - 1.41i)T - 5iT^{2} \) |
| 7 | \( 1 + 7iT^{2} \) |
| 11 | \( 1 + (-2.82 - 2.82i)T + 11iT^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 23iT^{2} \) |
| 29 | \( 1 + (7.07 - 7.07i)T - 29iT^{2} \) |
| 31 | \( 1 + (5.65 - 5.65i)T - 31iT^{2} \) |
| 37 | \( 1 + (-1.41 + 1.41i)T - 37iT^{2} \) |
| 41 | \( 1 + (-7.07 - 7.07i)T + 41iT^{2} \) |
| 43 | \( 1 + 12iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + (7.07 + 7.07i)T + 61iT^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 71iT^{2} \) |
| 73 | \( 1 + (-7.07 + 7.07i)T - 73iT^{2} \) |
| 79 | \( 1 + (-5.65 - 5.65i)T + 79iT^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + (9.89 - 9.89i)T - 97iT^{2} \) |
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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
−9.552002746007033629793805555068, −9.108853487719715451126966616775, −8.063329569668480837433520818345, −7.23587591883460382717590516392, −6.73762330629450599688885758334, −5.80467154241489835970827918340, −4.98838794897735425210161886000, −3.89336177111704867076803382650, −3.47989323980037405429060564971, −1.60428613811992023518741227950,
0.34428919154455157982916624769, 1.32624683027231639855781763160, 2.66313317371066589482155853989, 3.90869108617668724053920777837, 4.40296691061677148069820182139, 5.64567711561840772459996967512, 6.22999711527187383113308793123, 7.44527781545000573561576711323, 8.102436865409476266665150907627, 9.020592441950972739561132143517