L(s) = 1 | + (−1.42 − 1.42i)2-s + (−1.57 + 0.730i)3-s + 2.07i·4-s + (1.72 + 1.72i)5-s + (3.28 + 1.19i)6-s + (2.20 + 2.20i)7-s + (0.105 − 0.105i)8-s + (1.93 − 2.29i)9-s − 4.91i·10-s + (−1.95 + 1.95i)11-s + (−1.51 − 3.25i)12-s − 6.28i·14-s + (−3.96 − 1.44i)15-s + 3.84·16-s − 5.78·17-s + (−6.03 + 0.515i)18-s + ⋯ |
L(s) = 1 | + (−1.00 − 1.00i)2-s + (−0.906 + 0.421i)3-s + 1.03i·4-s + (0.770 + 0.770i)5-s + (1.34 + 0.489i)6-s + (0.831 + 0.831i)7-s + (0.0372 − 0.0372i)8-s + (0.644 − 0.764i)9-s − 1.55i·10-s + (−0.590 + 0.590i)11-s + (−0.437 − 0.940i)12-s − 1.67i·14-s + (−1.02 − 0.373i)15-s + 0.961·16-s − 1.40·17-s + (−1.42 + 0.121i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0145 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0145 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.328734 + 0.323969i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.328734 + 0.323969i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.57 - 0.730i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (1.42 + 1.42i)T + 2iT^{2} \) |
| 5 | \( 1 + (-1.72 - 1.72i)T + 5iT^{2} \) |
| 7 | \( 1 + (-2.20 - 2.20i)T + 7iT^{2} \) |
| 11 | \( 1 + (1.95 - 1.95i)T - 11iT^{2} \) |
| 17 | \( 1 + 5.78T + 17T^{2} \) |
| 19 | \( 1 + (1.06 - 1.06i)T - 19iT^{2} \) |
| 23 | \( 1 + 3.86T + 23T^{2} \) |
| 29 | \( 1 + 2.92iT - 29T^{2} \) |
| 31 | \( 1 + (3.56 - 3.56i)T - 31iT^{2} \) |
| 37 | \( 1 + (-2.83 - 2.83i)T + 37iT^{2} \) |
| 41 | \( 1 + (-4.79 - 4.79i)T + 41iT^{2} \) |
| 43 | \( 1 + 1.84iT - 43T^{2} \) |
| 47 | \( 1 + (-0.115 + 0.115i)T - 47iT^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 + (1.22 - 1.22i)T - 59iT^{2} \) |
| 61 | \( 1 + 5.39T + 61T^{2} \) |
| 67 | \( 1 + (9.65 - 9.65i)T - 67iT^{2} \) |
| 71 | \( 1 + (0.239 + 0.239i)T + 71iT^{2} \) |
| 73 | \( 1 + (8.54 + 8.54i)T + 73iT^{2} \) |
| 79 | \( 1 - 10.4T + 79T^{2} \) |
| 83 | \( 1 + (-2.83 - 2.83i)T + 83iT^{2} \) |
| 89 | \( 1 + (-3.00 + 3.00i)T - 89iT^{2} \) |
| 97 | \( 1 + (6.99 - 6.99i)T - 97iT^{2} \) |
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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
−10.80611093920414794995824449297, −10.48891496299737237913382887208, −9.598490055641155631611235548226, −8.884257893180446111841019160259, −7.75062963446785471150068928539, −6.40557452476354963541770319709, −5.59502183345908268390448849414, −4.43336770451677560977975615028, −2.64438751125992213647916930452, −1.79113811431017352252476764307,
0.42073744241789426135081391117, 1.78067541160611119220414621616, 4.39779052179380648687511603467, 5.42845400978754467046254237867, 6.17157994583765956409065302635, 7.15531129566005872930949563341, 7.899941935700200335353720479503, 8.733579313362687662688405364431, 9.636408120232575363680695167022, 10.67069578915175867253752703762