L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s − i·9-s + 11-s + (−0.866 + 0.500i)14-s + (0.500 + 0.866i)16-s + (−0.258 + 0.965i)18-s + (−0.965 − 0.258i)22-s + (−0.707 − 0.707i)23-s + (0.965 − 0.258i)28-s − 1.73·29-s + (−0.258 − 0.965i)32-s + (0.499 − 0.866i)36-s + (−0.707 + 0.707i)37-s + ⋯ |
L(s) = 1 | + (−0.965 − 0.258i)2-s + (0.866 + 0.499i)4-s + (0.707 − 0.707i)7-s + (−0.707 − 0.707i)8-s − i·9-s + 11-s + (−0.866 + 0.500i)14-s + (0.500 + 0.866i)16-s + (−0.258 + 0.965i)18-s + (−0.965 − 0.258i)22-s + (−0.707 − 0.707i)23-s + (0.965 − 0.258i)28-s − 1.73·29-s + (−0.258 − 0.965i)32-s + (0.499 − 0.866i)36-s + (−0.707 + 0.707i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.437 + 0.899i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7807588210\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7807588210\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.965 + 0.258i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-0.707 + 0.707i)T \) |
good | 3 | \( 1 + iT^{2} \) |
| 11 | \( 1 - T + T^{2} \) |
| 13 | \( 1 - iT^{2} \) |
| 17 | \( 1 - iT^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T + iT^{2} \) |
| 29 | \( 1 + 1.73T + T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (0.707 - 0.707i)T - iT^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 + (-1.22 + 1.22i)T - iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (-1.41 - 1.41i)T + iT^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + T^{2} \) |
| 67 | \( 1 + (-1.22 - 1.22i)T + iT^{2} \) |
| 71 | \( 1 + 1.73iT - T^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 - 1.73T + T^{2} \) |
| 83 | \( 1 + iT^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 - iT^{2} \) |
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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
−9.464295734647794446926253130682, −8.966381993999490312474479670923, −8.115897785799339333181986945100, −7.24954797013697223163412831513, −6.64900340183406250217496340989, −5.69972148102191602228657936648, −4.14826720011724330530796871942, −3.59180547247568932918178409059, −2.09774462166278239631058458330, −0.965022928397210697938829861752,
1.60644010234487088063850946054, 2.35406280460612877117047022117, 3.88068439778240306830719214553, 5.22699470549891823701584670596, 5.77398706145538517167772085548, 6.84472893533248034604240489843, 7.70384612717524716226263793053, 8.245826346231808969877528091148, 9.132964695617909487487879674066, 9.635754429380281529415529231330