L(s) = 1 | + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.777 − 0.974i)5-s + (−0.433 − 0.900i)8-s + (0.900 − 0.433i)9-s + (−1.21 − 0.277i)10-s + (−1.62 − 0.781i)13-s + (−0.900 − 0.433i)16-s + 0.445i·17-s + (0.433 − 0.900i)18-s + (−1.12 + 0.541i)20-s + (−0.123 + 0.541i)25-s + (−1.75 + 0.400i)26-s + (−0.974 + 0.222i)32-s + (0.277 + 0.347i)34-s + ⋯ |
L(s) = 1 | + (0.781 − 0.623i)2-s + (0.222 − 0.974i)4-s + (−0.777 − 0.974i)5-s + (−0.433 − 0.900i)8-s + (0.900 − 0.433i)9-s + (−1.21 − 0.277i)10-s + (−1.62 − 0.781i)13-s + (−0.900 − 0.433i)16-s + 0.445i·17-s + (0.433 − 0.900i)18-s + (−1.12 + 0.541i)20-s + (−0.123 + 0.541i)25-s + (−1.75 + 0.400i)26-s + (−0.974 + 0.222i)32-s + (0.277 + 0.347i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3364 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.988 + 0.151i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.399898013\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.399898013\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.781 + 0.623i)T \) |
| 29 | \( 1 \) |
good | 3 | \( 1 + (-0.900 + 0.433i)T^{2} \) |
| 5 | \( 1 + (0.777 + 0.974i)T + (-0.222 + 0.974i)T^{2} \) |
| 7 | \( 1 + (0.900 - 0.433i)T^{2} \) |
| 11 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 13 | \( 1 + (1.62 + 0.781i)T + (0.623 + 0.781i)T^{2} \) |
| 17 | \( 1 - 0.445iT - T^{2} \) |
| 19 | \( 1 + (-0.900 - 0.433i)T^{2} \) |
| 23 | \( 1 + (0.222 + 0.974i)T^{2} \) |
| 31 | \( 1 + (-0.222 + 0.974i)T^{2} \) |
| 37 | \( 1 + (-0.781 - 1.62i)T + (-0.623 + 0.781i)T^{2} \) |
| 41 | \( 1 + 1.80iT - T^{2} \) |
| 43 | \( 1 + (-0.222 - 0.974i)T^{2} \) |
| 47 | \( 1 + (0.623 + 0.781i)T^{2} \) |
| 53 | \( 1 + (0.277 + 0.347i)T + (-0.222 + 0.974i)T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 + (-0.433 + 0.0990i)T + (0.900 - 0.433i)T^{2} \) |
| 67 | \( 1 + (-0.623 + 0.781i)T^{2} \) |
| 71 | \( 1 + (-0.623 - 0.781i)T^{2} \) |
| 73 | \( 1 + (0.347 + 0.277i)T + (0.222 + 0.974i)T^{2} \) |
| 79 | \( 1 + (0.623 - 0.781i)T^{2} \) |
| 83 | \( 1 + (0.900 + 0.433i)T^{2} \) |
| 89 | \( 1 + (-0.974 + 0.777i)T + (0.222 - 0.974i)T^{2} \) |
| 97 | \( 1 + (-1.75 - 0.400i)T + (0.900 + 0.433i)T^{2} \) |
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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
−8.455996829108700748252220084591, −7.63012741344669249531042534267, −6.95815940780268435379938797305, −5.99833345537054087146380168987, −5.00923945062221528441815391135, −4.62411000072276766131942932959, −3.85231825416908246018916041636, −2.95349529792161803584043462484, −1.79283196647504301411075781208, −0.62012619899130350259905919859,
2.13873976212417495930508196061, 2.90521461472348975578268550059, 3.88806021284573918515411790184, 4.56088133526715135003814067616, 5.18508475504228907662276699962, 6.40050571593513447530939031035, 6.93317902872135782286579677957, 7.61204486624878628530896040213, 7.81599535945666756556402624590, 9.114742232830911301822201843221