L(s) = 1 | + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (−11 + 19.0i)5-s + 7.99·8-s + (−22 − 38.1i)10-s + (13 + 22.5i)11-s + 54·13-s + (−8 + 13.8i)16-s + (−37 − 64.0i)17-s + (58 − 100. i)19-s + 88·20-s − 51.9·22-s + (−29 + 50.2i)23-s + (−179.5 − 310. i)25-s + (−54 + 93.5i)26-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.983 + 1.70i)5-s + 0.353·8-s + (−0.695 − 1.20i)10-s + (0.356 + 0.617i)11-s + 1.15·13-s + (−0.125 + 0.216i)16-s + (−0.527 − 0.914i)17-s + (0.700 − 1.21i)19-s + 0.983·20-s − 0.503·22-s + (−0.262 + 0.455i)23-s + (−1.43 − 2.48i)25-s + (−0.407 + 0.705i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.968 - 0.250i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(0.9782505264\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9782505264\) |
\(L(\frac{5}{2})\) |
|
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 | \( 1 + (11 - 19.0i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-13 - 22.5i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 54T + 2.19e3T^{2} \) |
| 17 | \( 1 + (37 + 64.0i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-58 + 100. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (29 - 50.2i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 208T + 2.43e4T^{2} \) |
| 31 | \( 1 + (126 + 218. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (25 - 43.3i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 126T + 6.89e4T^{2} \) |
| 43 | \( 1 - 164T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-222 + 384. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-6 - 10.3i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (62 + 107. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (81 - 140. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-430 - 744. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 238T + 3.57e5T^{2} \) |
| 73 | \( 1 + (73 + 126. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-492 + 852. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 656T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-477 + 826. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 526T + 9.12e5T^{2} \) |
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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
−9.679307336427819399674734133445, −8.948596996931048444671945232893, −7.74052868467218168454395702554, −7.25703577868893716890518969158, −6.65411016743907223561366470271, −5.67838750235509935307533007505, −4.27028959715266063576473322013, −3.46918096519184009751037377914, −2.26748573469788549057938807006, −0.39013114555861141962453498791,
0.887657582814750974038995961043, 1.63873869919588307507488588616, 3.68054625990379606434947005291, 3.92162006888989307190914822155, 5.17091914651107772065302163293, 6.09380494144467506112793324923, 7.63091023140168446810119532475, 8.248814367498077897165265047481, 8.878316133808982505581640176564, 9.432825958910248242204933853110