L(s) = 1 | + 5.58i·2-s − 5.19i·3-s − 15.1·4-s − 19.0i·5-s + 29.0·6-s + 39.7i·7-s + 4.74i·8-s − 27·9-s + 106.·10-s − 26.8i·11-s + 78.7i·12-s − 248. i·13-s − 221.·14-s − 98.9·15-s − 268.·16-s + 281.·17-s + ⋯ |
L(s) = 1 | + 1.39i·2-s − 0.577i·3-s − 0.946·4-s − 0.761i·5-s + 0.805·6-s + 0.810i·7-s + 0.0741i·8-s − 0.333·9-s + 1.06·10-s − 0.221i·11-s + 0.546i·12-s − 1.46i·13-s − 1.13·14-s − 0.439·15-s − 1.05·16-s + 0.974·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.655 - 0.754i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 201 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.655 - 0.754i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(1.920898877\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.920898877\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 5.19iT \) |
| 67 | \( 1 + (-3.38e3 - 2.94e3i)T \) |
good | 2 | \( 1 - 5.58iT - 16T^{2} \) |
| 5 | \( 1 + 19.0iT - 625T^{2} \) |
| 7 | \( 1 - 39.7iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 26.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 248. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 281.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 293.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 651.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 34.6T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.29e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 - 2.47e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 380. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.34e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 1.21e3T + 4.87e6T^{2} \) |
| 53 | \( 1 + 2.84e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 + 5.20e3T + 1.21e7T^{2} \) |
| 61 | \( 1 - 112. iT - 1.38e7T^{2} \) |
| 71 | \( 1 - 8.42e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 8.33e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 4.89e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 3.33e3T + 4.74e7T^{2} \) |
| 89 | \( 1 + 4.74e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 8.42e3iT - 8.85e7T^{2} \) |
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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
−12.27907240492270074450770784943, −11.04474304913246806457866327117, −9.466156087570529230044106897127, −8.469547518363564334336504267201, −7.894266299024246571483958356759, −6.79739831687382976431406053267, −5.53364027163110474777472240600, −5.19447902910706793427625336627, −2.93334904186465758412699866557, −0.905818661724032156598056152495,
1.07154554523730078703027990100, 2.66145730271801342500748732567, 3.69449962943420024318380728843, 4.66482493465492834403892126195, 6.50997957038108077463837187558, 7.57857282915469657952156252376, 9.375806112844494539909326759187, 9.786062599839434019401961741524, 10.94897030991133737630783847684, 11.25997698662269556750859512029