L(s) = 1 | + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.707 − 1.70i)11-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (0.707 + 1.70i)29-s + (−0.707 − 0.292i)37-s + (−0.292 + 0.707i)43-s + 1.00i·49-s + (0.292 − 0.707i)53-s − 1.00·63-s + (0.292 + 0.707i)67-s + (1.70 − 0.707i)77-s − 1.41i·79-s − 1.00i·81-s + ⋯ |
L(s) = 1 | + (0.707 + 0.707i)7-s + (−0.707 + 0.707i)9-s + (0.707 − 1.70i)11-s + (1 − i)23-s + (0.707 + 0.707i)25-s + (0.707 + 1.70i)29-s + (−0.707 − 0.292i)37-s + (−0.292 + 0.707i)43-s + 1.00i·49-s + (0.292 − 0.707i)53-s − 1.00·63-s + (0.292 + 0.707i)67-s + (1.70 − 0.707i)77-s − 1.41i·79-s − 1.00i·81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.195i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.239284255\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.239284255\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 5 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 11 | \( 1 + (-0.707 + 1.70i)T + (-0.707 - 0.707i)T^{2} \) |
| 13 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 17 | \( 1 + T^{2} \) |
| 19 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (-1 + i)T - iT^{2} \) |
| 29 | \( 1 + (-0.707 - 1.70i)T + (-0.707 + 0.707i)T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 41 | \( 1 + iT^{2} \) |
| 43 | \( 1 + (0.292 - 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 61 | \( 1 + (0.707 - 0.707i)T^{2} \) |
| 67 | \( 1 + (-0.292 - 0.707i)T + (-0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + iT^{2} \) |
| 73 | \( 1 + iT^{2} \) |
| 79 | \( 1 + 1.41iT - T^{2} \) |
| 83 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 89 | \( 1 - iT^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.113887543703897793157731824346, −8.662143040434386839256474904162, −8.261420061511199060990440630327, −7.08711334295617982388325631531, −6.21576793495913288672666938555, −5.37455811743892281392160070934, −4.82380898619976756580846364012, −3.40043955131161434587863848562, −2.69214100945312238063006563296, −1.30402428085307831129897323018,
1.21016810437083424934966732154, 2.42588627887330290458188613574, 3.70794689649359819933394860728, 4.47813848181856758167666029695, 5.25395544587937548126499269955, 6.45542394977899089445833295075, 7.03167469430514836310008393642, 7.82568194844155562066348528742, 8.697089358431891805090083378100, 9.479559546712204135502919089780