| L(s) = 1 | + (−1 + 1.73i)2-s + (−1.99 − 3.46i)4-s + (4 − 6.92i)5-s + 7.99·8-s + (7.99 + 13.8i)10-s + (20 + 34.6i)11-s + 4·13-s + (−8 + 13.8i)16-s + (−42 − 72.7i)17-s + (−74 + 128. i)19-s − 31.9·20-s − 80·22-s + (42 − 72.7i)23-s + (30.5 + 52.8i)25-s + (−4 + 6.92i)26-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.357 − 0.619i)5-s + 0.353·8-s + (0.252 + 0.438i)10-s + (0.548 + 0.949i)11-s + 0.0853·13-s + (−0.125 + 0.216i)16-s + (−0.599 − 1.03i)17-s + (−0.893 + 1.54i)19-s − 0.357·20-s − 0.775·22-s + (0.380 − 0.659i)23-s + (0.244 + 0.422i)25-s + (−0.0301 + 0.0522i)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
| \(L(2)\) |
\(\approx\) |
\(1.262254232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.262254232\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + (-4 + 6.92i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (-20 - 34.6i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 - 4T + 2.19e3T^{2} \) |
| 17 | \( 1 + (42 + 72.7i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (74 - 128. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (-42 + 72.7i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 + 58T + 2.43e4T^{2} \) |
| 31 | \( 1 + (-68 - 117. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (-111 + 192. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 + 420T + 6.89e4T^{2} \) |
| 43 | \( 1 + 164T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-244 + 422. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-239 - 413. i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (-274 - 474. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (346 - 599. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-454 - 786. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 - 524T + 3.57e5T^{2} \) |
| 73 | \( 1 + (220 + 381. i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (608 - 1.05e3i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 - 684T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-302 + 523. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 + 832T + 9.12e5T^{2} \) |
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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
−9.910075997239317237751294079883, −8.965411909394660092936402786834, −8.552416923603640979856184080166, −7.34338592388613631723495004630, −6.72557698150347431004615973683, −5.69226922168415090878288138083, −4.84168424903329939716650601645, −3.94561779717954434556179207480, −2.22051265698941903484376616446, −1.12280294265039273695632535460,
0.40372768027252831420413483416, 1.80638605263437487070091983098, 2.85382632787840991472545488910, 3.79843187422710614307505161963, 4.90822406959509140830949012500, 6.28986634520893858265906278529, 6.72590039828169247260887310317, 8.069712964044986680870957581835, 8.746484285154261989225518921380, 9.520824818266280354133375905641
