| L(s) = 1 | − 1.79·3-s + 0.791i·5-s − 2·7-s + 0.208·9-s + 0.791·11-s − 3.79i·13-s − 1.41i·15-s − 7.58i·17-s + 1.58i·19-s + 3.58·21-s + 3.79i·23-s + 4.37·25-s + 5.00·27-s + 3.79i·29-s + 8.37i·31-s + ⋯ |
| L(s) = 1 | − 1.03·3-s + 0.353i·5-s − 0.755·7-s + 0.0695·9-s + 0.238·11-s − 1.05i·13-s − 0.365i·15-s − 1.83i·17-s + 0.363i·19-s + 0.781·21-s + 0.790i·23-s + 0.874·25-s + 0.962·27-s + 0.704i·29-s + 1.50i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.657 - 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3144978821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3144978821\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 + (4 + 4.58i)T \) |
| good | 3 | \( 1 + 1.79T + 3T^{2} \) |
| 5 | \( 1 - 0.791iT - 5T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 0.791T + 11T^{2} \) |
| 13 | \( 1 + 3.79iT - 13T^{2} \) |
| 17 | \( 1 + 7.58iT - 17T^{2} \) |
| 19 | \( 1 - 1.58iT - 19T^{2} \) |
| 23 | \( 1 - 3.79iT - 23T^{2} \) |
| 29 | \( 1 - 3.79iT - 29T^{2} \) |
| 31 | \( 1 - 8.37iT - 31T^{2} \) |
| 41 | \( 1 - 9.79T + 41T^{2} \) |
| 43 | \( 1 + 6iT - 43T^{2} \) |
| 47 | \( 1 + 7.58T + 47T^{2} \) |
| 53 | \( 1 - 1.58T + 53T^{2} \) |
| 59 | \( 1 - 1.58iT - 59T^{2} \) |
| 61 | \( 1 - 12.7iT - 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 + 9.16T + 71T^{2} \) |
| 73 | \( 1 + 4.37T + 73T^{2} \) |
| 79 | \( 1 + 8.20iT - 79T^{2} \) |
| 83 | \( 1 + 15.1T + 83T^{2} \) |
| 89 | \( 1 + 6iT - 89T^{2} \) |
| 97 | \( 1 - 13.5iT - 97T^{2} \) |
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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
−9.274628977285619658195847610630, −8.635562691457647464926748471809, −7.32360981832508907413864774207, −7.01940088051621009308746880831, −6.03325097743574270061670030505, −5.43678790245032273724440981213, −4.74137803410842131042865470272, −3.34082151828458863412535332824, −2.82197321377527603898724064119, −1.03876516448623958378482270295,
0.14818706024205912780459585894, 1.51991019672301932102880842952, 2.83578311042653996308447391439, 4.08387066834078295097238470514, 4.64509546806035514743474672592, 5.77245418628657951137056206766, 6.33272196991110120972540135389, 6.77010669713409577997910689610, 8.026522380050182768680990946860, 8.715551177699091573902785611356
