L(s) = 1 | + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (−0.349 − 0.172i)5-s + (0.923 − 0.382i)8-s + i·9-s + (−0.369 − 0.125i)10-s + (−0.125 + 0.630i)13-s + (0.866 − 0.5i)16-s + (0.108 − 1.65i)17-s + (0.130 + 0.991i)18-s + (−0.382 − 0.0761i)20-s + (−0.516 − 0.672i)25-s + (−0.0420 + 0.641i)26-s + (−1.85 + 0.369i)29-s + (0.793 − 0.608i)32-s + ⋯ |
L(s) = 1 | + (0.991 − 0.130i)2-s + (0.965 − 0.258i)4-s + (−0.349 − 0.172i)5-s + (0.923 − 0.382i)8-s + i·9-s + (−0.369 − 0.125i)10-s + (−0.125 + 0.630i)13-s + (0.866 − 0.5i)16-s + (0.108 − 1.65i)17-s + (0.130 + 0.991i)18-s + (−0.382 − 0.0761i)20-s + (−0.516 − 0.672i)25-s + (−0.0420 + 0.641i)26-s + (−1.85 + 0.369i)29-s + (0.793 − 0.608i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 772 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.978 + 0.204i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.647167030\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.647167030\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.991 + 0.130i)T \) |
| 193 | \( 1 + (-0.707 - 0.707i)T \) |
good | 3 | \( 1 - iT^{2} \) |
| 5 | \( 1 + (0.349 + 0.172i)T + (0.608 + 0.793i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (0.382 + 0.923i)T^{2} \) |
| 13 | \( 1 + (0.125 - 0.630i)T + (-0.923 - 0.382i)T^{2} \) |
| 17 | \( 1 + (-0.108 + 1.65i)T + (-0.991 - 0.130i)T^{2} \) |
| 19 | \( 1 + (-0.608 - 0.793i)T^{2} \) |
| 23 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 29 | \( 1 + (1.85 - 0.369i)T + (0.923 - 0.382i)T^{2} \) |
| 31 | \( 1 + (-0.965 + 0.258i)T^{2} \) |
| 37 | \( 1 + (0.583 - 0.665i)T + (-0.130 - 0.991i)T^{2} \) |
| 41 | \( 1 + (1.50 + 0.0983i)T + (0.991 + 0.130i)T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + (-0.130 - 0.991i)T^{2} \) |
| 53 | \( 1 + (-1.50 - 1.31i)T + (0.130 + 0.991i)T^{2} \) |
| 59 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 61 | \( 1 + (1.18 - 0.583i)T + (0.608 - 0.793i)T^{2} \) |
| 67 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 71 | \( 1 + (-0.382 + 0.923i)T^{2} \) |
| 73 | \( 1 + (-0.0862 - 0.0983i)T + (-0.130 + 0.991i)T^{2} \) |
| 79 | \( 1 + (0.608 - 0.793i)T^{2} \) |
| 83 | \( 1 + (-0.258 + 0.965i)T^{2} \) |
| 89 | \( 1 + (0.382 + 1.92i)T + (-0.923 + 0.382i)T^{2} \) |
| 97 | \( 1 + (-0.513 - 0.0675i)T + (0.965 + 0.258i)T^{2} \) |
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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
−10.70085541386444839569093388497, −9.851325931656069275692730653310, −8.769932620879296922752366365681, −7.53289972853495834971682040044, −7.13641703065656690712189782773, −5.83672368587621120841898220641, −4.98712762436456591266089079920, −4.24748493943247380572428918156, −3.01860628534416485045486503224, −1.89005802164656225508777371854,
1.90924306303309231507689577665, 3.53405666006800496943079287722, 3.81868230976407912583436986179, 5.30221908640517079152443423893, 6.02254488355715884306631878604, 6.94043664043590038848562287548, 7.77297984791195316789458147068, 8.679963264952983189900844088611, 9.899826814389973386788085688870, 10.73331643169922461569523221900