L(s) = 1 | + (0.781 + 0.623i)2-s + (0.222 + 0.974i)4-s + (−1.04 − 0.505i)5-s + (−2.62 + 0.311i)7-s + (−0.433 + 0.900i)8-s + (−0.505 − 1.04i)10-s + (2.62 + 2.09i)11-s + (2.75 + 2.19i)13-s + (−2.24 − 1.39i)14-s + (−0.900 + 0.433i)16-s + (0.165 − 0.722i)17-s + 6.51i·19-s + (0.259 − 1.13i)20-s + (0.747 + 3.27i)22-s + (−8.22 + 1.87i)23-s + ⋯ |
L(s) = 1 | + (0.552 + 0.440i)2-s + (0.111 + 0.487i)4-s + (−0.469 − 0.225i)5-s + (−0.993 + 0.117i)7-s + (−0.153 + 0.318i)8-s + (−0.159 − 0.331i)10-s + (0.791 + 0.631i)11-s + (0.763 + 0.608i)13-s + (−0.600 − 0.372i)14-s + (−0.225 + 0.108i)16-s + (0.0400 − 0.175i)17-s + 1.49i·19-s + (0.0579 − 0.253i)20-s + (0.159 + 0.698i)22-s + (−1.71 + 0.391i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.662 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.559697 + 1.24125i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.559697 + 1.24125i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.781 - 0.623i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.62 - 0.311i)T \) |
good | 5 | \( 1 + (1.04 + 0.505i)T + (3.11 + 3.90i)T^{2} \) |
| 11 | \( 1 + (-2.62 - 2.09i)T + (2.44 + 10.7i)T^{2} \) |
| 13 | \( 1 + (-2.75 - 2.19i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.165 + 0.722i)T + (-15.3 - 7.37i)T^{2} \) |
| 19 | \( 1 - 6.51iT - 19T^{2} \) |
| 23 | \( 1 + (8.22 - 1.87i)T + (20.7 - 9.97i)T^{2} \) |
| 29 | \( 1 + (2.13 + 0.488i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 - 8.50iT - 31T^{2} \) |
| 37 | \( 1 + (2.22 - 9.73i)T + (-33.3 - 16.0i)T^{2} \) |
| 41 | \( 1 + (0.840 + 0.404i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (10.1 - 4.88i)T + (26.8 - 33.6i)T^{2} \) |
| 47 | \( 1 + (-6.85 + 8.59i)T + (-10.4 - 45.8i)T^{2} \) |
| 53 | \( 1 + (5.82 - 1.33i)T + (47.7 - 22.9i)T^{2} \) |
| 59 | \( 1 + (-12.5 + 6.05i)T + (36.7 - 46.1i)T^{2} \) |
| 61 | \( 1 + (-4.31 - 0.985i)T + (54.9 + 26.4i)T^{2} \) |
| 67 | \( 1 - 7.58T + 67T^{2} \) |
| 71 | \( 1 + (-2.95 + 0.673i)T + (63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-11.1 + 8.90i)T + (16.2 - 71.1i)T^{2} \) |
| 79 | \( 1 - 10.1T + 79T^{2} \) |
| 83 | \( 1 + (-1.64 - 2.06i)T + (-18.4 + 80.9i)T^{2} \) |
| 89 | \( 1 + (3.51 + 4.40i)T + (-19.8 + 86.7i)T^{2} \) |
| 97 | \( 1 - 3.07iT - 97T^{2} \) |
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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
−10.24145478035543876202690811890, −9.659651902359739706341170161838, −8.570125662527210761756859899285, −7.923565782191183704910691036956, −6.69176323564131527894110791002, −6.34270006585474854509260141739, −5.19989373008751082754002386998, −3.92643633631466120120066631974, −3.58229375961596486236698643289, −1.82565506422182734814864912176,
0.53656540818066777515053168386, 2.37155907556002377699790056286, 3.64170580144655040235426432374, 3.94924082096475677420260642809, 5.53193514517539415272989486595, 6.24425264574225131235062504864, 7.06967951734521561380544431643, 8.180199523547730557422053895870, 9.172993897559201587018053687061, 9.888349954722284538386371524946