L(s) = 1 | + (−1 − 1.73i)2-s + (−1.99 + 3.46i)4-s + (7.61 + 13.1i)5-s + 7.99·8-s + (15.2 − 26.3i)10-s + (−1 + 1.73i)11-s + 30.4·13-s + (−8 − 13.8i)16-s + (−22.8 + 39.5i)17-s + (76.1 + 131. i)19-s − 60.9·20-s + 3.99·22-s + (−15 − 25.9i)23-s + (−53.5 + 92.6i)25-s + (−30.4 − 52.7i)26-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.249 + 0.433i)4-s + (0.681 + 1.17i)5-s + 0.353·8-s + (0.481 − 0.834i)10-s + (−0.0274 + 0.0474i)11-s + 0.649·13-s + (−0.125 − 0.216i)16-s + (−0.325 + 0.564i)17-s + (0.919 + 1.59i)19-s − 0.681·20-s + 0.0387·22-s + (−0.135 − 0.235i)23-s + (−0.427 + 0.741i)25-s + (−0.229 − 0.397i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.797763847\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.797763847\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (1 + 1.73i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-7.61 - 13.1i)T + (-62.5 + 108. i)T^{2} \) |
| 11 | \( 1 + (1 - 1.73i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 - 30.4T + 2.19e3T^{2} \) |
| 17 | \( 1 + (22.8 - 39.5i)T + (-2.45e3 - 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-76.1 - 131. i)T + (-3.42e3 + 5.94e3i)T^{2} \) |
| 23 | \( 1 + (15 + 25.9i)T + (-6.08e3 + 1.05e4i)T^{2} \) |
| 29 | \( 1 - 212T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-106. + 184. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + (123 + 213. i)T + (-2.53e4 + 4.38e4i)T^{2} \) |
| 41 | \( 1 - 319.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 284T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-30.4 - 52.7i)T + (-5.19e4 + 8.99e4i)T^{2} \) |
| 53 | \( 1 + (274 - 474. i)T + (-7.44e4 - 1.28e5i)T^{2} \) |
| 59 | \( 1 + (335. - 580. i)T + (-1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (258. + 448. i)T + (-1.13e5 + 1.96e5i)T^{2} \) |
| 67 | \( 1 + (326 - 564. i)T + (-1.50e5 - 2.60e5i)T^{2} \) |
| 71 | \( 1 - 770T + 3.57e5T^{2} \) |
| 73 | \( 1 + (487. - 844. i)T + (-1.94e5 - 3.36e5i)T^{2} \) |
| 79 | \( 1 + (236 + 408. i)T + (-2.46e5 + 4.26e5i)T^{2} \) |
| 83 | \( 1 + 182.T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-357. - 619. i)T + (-3.52e5 + 6.10e5i)T^{2} \) |
| 97 | \( 1 + 304.T + 9.12e5T^{2} \) |
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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
−10.16968836604202739733179358150, −9.284653337075926306567966595169, −8.269835351662694288768657879687, −7.47386358585205381603305183745, −6.37231445600240265059460091363, −5.78973150129711513649653004042, −4.26379832998782045986602295769, −3.27303342996702108862046147078, −2.36661247360830379367372947633, −1.25859933387705786838903877597,
0.58051030137407099959542647424, 1.50967372765508650199969297650, 3.04238548924186821063007445591, 4.75241188660656020193570739131, 5.04628470369222934014733936784, 6.20117087815480231871551767231, 6.94418379319620600555524861359, 8.088723849369965719541666400918, 8.818062998700698037827012529475, 9.343841336080027483643874717545