L(s) = 1 | + (1.52 − 0.347i)5-s + (0.781 − 0.623i)9-s + (−1.40 − 1.12i)13-s + (0.752 + 0.752i)17-s + (1.30 − 0.626i)25-s + (−0.781 − 0.623i)29-s + (−0.656 − 0.0739i)37-s + (−1.33 + 1.33i)41-s + (0.974 − 1.22i)45-s + (−0.623 − 0.781i)49-s + (0.433 + 1.90i)53-s + (0.559 + 1.59i)61-s + (−2.53 − 1.22i)65-s + (0.900 + 1.43i)73-s + (0.222 − 0.974i)81-s + ⋯ |
L(s) = 1 | + (1.52 − 0.347i)5-s + (0.781 − 0.623i)9-s + (−1.40 − 1.12i)13-s + (0.752 + 0.752i)17-s + (1.30 − 0.626i)25-s + (−0.781 − 0.623i)29-s + (−0.656 − 0.0739i)37-s + (−1.33 + 1.33i)41-s + (0.974 − 1.22i)45-s + (−0.623 − 0.781i)49-s + (0.433 + 1.90i)53-s + (0.559 + 1.59i)61-s + (−2.53 − 1.22i)65-s + (0.900 + 1.43i)73-s + (0.222 − 0.974i)81-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.857 + 0.514i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.496542999\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.496542999\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 29 | \( 1 + (0.781 + 0.623i)T \) |
good | 3 | \( 1 + (-0.781 + 0.623i)T^{2} \) |
| 5 | \( 1 + (-1.52 + 0.347i)T + (0.900 - 0.433i)T^{2} \) |
| 7 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 11 | \( 1 + (0.974 - 0.222i)T^{2} \) |
| 13 | \( 1 + (1.40 + 1.12i)T + (0.222 + 0.974i)T^{2} \) |
| 17 | \( 1 + (-0.752 - 0.752i)T + iT^{2} \) |
| 19 | \( 1 + (0.781 + 0.623i)T^{2} \) |
| 23 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 31 | \( 1 + (-0.433 - 0.900i)T^{2} \) |
| 37 | \( 1 + (0.656 + 0.0739i)T + (0.974 + 0.222i)T^{2} \) |
| 41 | \( 1 + (1.33 - 1.33i)T - iT^{2} \) |
| 43 | \( 1 + (0.433 - 0.900i)T^{2} \) |
| 47 | \( 1 + (-0.974 + 0.222i)T^{2} \) |
| 53 | \( 1 + (-0.433 - 1.90i)T + (-0.900 + 0.433i)T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-0.559 - 1.59i)T + (-0.781 + 0.623i)T^{2} \) |
| 67 | \( 1 + (0.222 - 0.974i)T^{2} \) |
| 71 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 73 | \( 1 + (-0.900 - 1.43i)T + (-0.433 + 0.900i)T^{2} \) |
| 79 | \( 1 + (-0.974 - 0.222i)T^{2} \) |
| 83 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 89 | \( 1 + (-1.05 + 1.68i)T + (-0.433 - 0.900i)T^{2} \) |
| 97 | \( 1 + (0.218 - 0.623i)T + (-0.781 - 0.623i)T^{2} \) |
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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
−9.602772037862711844058266652093, −8.715661226837158061175401252598, −7.75555853963149264991444801350, −6.96062433769143550892691903362, −6.01279891676909728253590419640, −5.44923525998251875558629041007, −4.61661126904549965015475251594, −3.37801666994883554862807514778, −2.29088214902001502614399833689, −1.27135742673913771491527722652,
1.77824934573855708675068082446, 2.27448776592897472172498474750, 3.56331741495776058511837354229, 5.07552135731816097118394820481, 5.12700631925494245668625718555, 6.49237372493685082391101846581, 7.00965529726659132643101886825, 7.74931283995296260648297458852, 9.019397399257344298115345763767, 9.632560319087923702526564682809