| L(s) = 1 | + i·2-s + 3.03i·3-s − 4-s − 3.03·6-s + 2.46i·7-s − i·8-s − 6.19·9-s + 0.728·11-s − 3.03i·12-s + 6.23i·13-s − 2.46·14-s + 16-s + 0.563i·17-s − 6.19i·18-s + 19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 1.75i·3-s − 0.5·4-s − 1.23·6-s + 0.933i·7-s − 0.353i·8-s − 2.06·9-s + 0.219·11-s − 0.875i·12-s + 1.72i·13-s − 0.660·14-s + 0.250·16-s + 0.136i·17-s − 1.46i·18-s + 0.229·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 950 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.599132 - 0.969417i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.599132 - 0.969417i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 3.03iT - 3T^{2} \) |
| 7 | \( 1 - 2.46iT - 7T^{2} \) |
| 11 | \( 1 - 0.728T + 11T^{2} \) |
| 13 | \( 1 - 6.23iT - 13T^{2} \) |
| 17 | \( 1 - 0.563iT - 17T^{2} \) |
| 23 | \( 1 + 4.63iT - 23T^{2} \) |
| 29 | \( 1 + 10.2T + 29T^{2} \) |
| 31 | \( 1 - 6.06T + 31T^{2} \) |
| 37 | \( 1 + 5.72iT - 37T^{2} \) |
| 41 | \( 1 - 4.79T + 41T^{2} \) |
| 43 | \( 1 + 8.06iT - 43T^{2} \) |
| 47 | \( 1 - 8.12iT - 47T^{2} \) |
| 53 | \( 1 - 1.53iT - 53T^{2} \) |
| 59 | \( 1 - 5.76T + 59T^{2} \) |
| 61 | \( 1 - 10.9T + 61T^{2} \) |
| 67 | \( 1 - 12.9iT - 67T^{2} \) |
| 71 | \( 1 + 4.39T + 71T^{2} \) |
| 73 | \( 1 - 4.09iT - 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 + 7.85iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 11.0iT - 97T^{2} \) |
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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
−10.37772803262404080473937016563, −9.492811624209865164620765616155, −9.050089046060552111810945216321, −8.500693345445627374180009519947, −7.15168726712593702235939717049, −6.05540930528989514022884749402, −5.39537682621277548494733873925, −4.39348245550699026646732106531, −3.86774574820099920160136304989, −2.42777354679122028722842091150,
0.55640147237257796701151293681, 1.47669690046997311806703958247, 2.74498952232078345304881435295, 3.67831137126918413856069349132, 5.22960580558928601152261085071, 6.07858436103319907958096065958, 7.16663547524415812857534170425, 7.74242295226464328730948966670, 8.374406721507450405134876753779, 9.584365608553994984386940221385
