L(s) = 1 | − 2.82i·2-s + (5 − 1.41i)3-s − 8.00·4-s + (−4.00 − 14.1i)6-s + 22.6i·8-s + (23 − 14.1i)9-s + 70.7i·11-s + (−40.0 + 11.3i)12-s + 64.0·16-s − 107. i·17-s + (−40.0 − 65.0i)18-s − 106·19-s + 200.·22-s + (32.0 + 113. i)24-s − 125·25-s + ⋯ |
L(s) = 1 | − 0.999i·2-s + (0.962 − 0.272i)3-s − 1.00·4-s + (−0.272 − 0.962i)6-s + 1.00i·8-s + (0.851 − 0.523i)9-s + 1.93i·11-s + (−0.962 + 0.272i)12-s + 1.00·16-s − 1.53i·17-s + (−0.523 − 0.851i)18-s − 1.27·19-s + 1.93·22-s + (0.272 + 0.962i)24-s − 25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.272 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.05837 - 0.800543i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.05837 - 0.800543i\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + 2.82iT \) |
| 3 | \( 1 + (-5 + 1.41i)T \) |
good | 5 | \( 1 + 125T^{2} \) |
| 7 | \( 1 - 343T^{2} \) |
| 11 | \( 1 - 70.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 2.19e3T^{2} \) |
| 17 | \( 1 + 107. iT - 4.91e3T^{2} \) |
| 19 | \( 1 + 106T + 6.85e3T^{2} \) |
| 23 | \( 1 + 1.21e4T^{2} \) |
| 29 | \( 1 + 2.43e4T^{2} \) |
| 31 | \( 1 - 2.97e4T^{2} \) |
| 37 | \( 1 - 5.06e4T^{2} \) |
| 41 | \( 1 + 56.5iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 290T + 7.95e4T^{2} \) |
| 47 | \( 1 + 1.03e5T^{2} \) |
| 53 | \( 1 + 1.48e5T^{2} \) |
| 59 | \( 1 - 325. iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 2.26e5T^{2} \) |
| 67 | \( 1 + 70T + 3.00e5T^{2} \) |
| 71 | \( 1 + 3.57e5T^{2} \) |
| 73 | \( 1 + 430T + 3.89e5T^{2} \) |
| 79 | \( 1 - 4.93e5T^{2} \) |
| 83 | \( 1 - 681. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.32e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 1.91e3T + 9.12e5T^{2} \) |
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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
−17.49537251847867538030445534778, −15.34802142870530138950796854847, −14.23141439033694605645255625899, −13.01653052415009321306624471507, −12.01523309172515279372958118806, −10.12289544376962637171298744557, −9.108404416454209027254155882530, −7.44201540410787076520053969815, −4.37526749996133579850317243106, −2.26027663966447729927884215960,
3.85775137198782991165286996682, 6.07442499956795941981547089926, 8.045852476982021650881049614039, 8.873267286634214673132337974018, 10.54909112750273714796651660759, 12.99333616926732856934058342174, 13.97190630998819148280329432401, 15.02070220758083529490572384490, 16.09487684216355360453776254408, 17.17707662552019832708411484596