| L(s) = 1 | + i·3-s + (−0.181 − 2.22i)5-s − 0.189i·7-s − 9-s + 4.33·11-s − 4.67i·13-s + (2.22 − 0.181i)15-s + 0.0397i·17-s − 7.19·19-s + 0.189·21-s − i·23-s + (−4.93 + 0.810i)25-s − i·27-s − 6.49·29-s + 9.17·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + (−0.0812 − 0.996i)5-s − 0.0717i·7-s − 0.333·9-s + 1.30·11-s − 1.29i·13-s + (0.575 − 0.0469i)15-s + 0.00963i·17-s − 1.65·19-s + 0.0414·21-s − 0.208i·23-s + (−0.986 + 0.162i)25-s − 0.192i·27-s − 1.20·29-s + 1.64·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0812 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0812 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.318674464\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.318674464\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 + (0.181 + 2.22i)T \) |
| 23 | \( 1 + iT \) |
| good | 7 | \( 1 + 0.189iT - 7T^{2} \) |
| 11 | \( 1 - 4.33T + 11T^{2} \) |
| 13 | \( 1 + 4.67iT - 13T^{2} \) |
| 17 | \( 1 - 0.0397iT - 17T^{2} \) |
| 19 | \( 1 + 7.19T + 19T^{2} \) |
| 29 | \( 1 + 6.49T + 29T^{2} \) |
| 31 | \( 1 - 9.17T + 31T^{2} \) |
| 37 | \( 1 + 3.03iT - 37T^{2} \) |
| 41 | \( 1 + 8.69T + 41T^{2} \) |
| 43 | \( 1 + 7.83iT - 43T^{2} \) |
| 47 | \( 1 + 5.03iT - 47T^{2} \) |
| 53 | \( 1 + 6.96iT - 53T^{2} \) |
| 59 | \( 1 + 4.64T + 59T^{2} \) |
| 61 | \( 1 - 7.86T + 61T^{2} \) |
| 67 | \( 1 + 2.52iT - 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 + 6.29iT - 73T^{2} \) |
| 79 | \( 1 - 9.82T + 79T^{2} \) |
| 83 | \( 1 + 0.417iT - 83T^{2} \) |
| 89 | \( 1 - 3.92T + 89T^{2} \) |
| 97 | \( 1 + 0.0584iT - 97T^{2} \) |
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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
−9.316639530095199265417777733130, −8.599995531522785787935637287834, −8.114892777619142425091976188961, −6.86043385359021847771674022665, −5.94858660504313689032932796058, −5.11947434517301769643696324859, −4.21964704449920823402237363560, −3.54689489710768618249499962427, −2.01424733107836405543617490388, −0.54101903885313742496317276885,
1.55480172581313373824791271197, 2.51793462210660782982282074051, 3.74602649878921554320296592464, 4.50243112682224922980814680407, 6.08485001784640318379589362517, 6.52919318508949267131577054154, 7.10870092371497683089106868733, 8.133497120237896862275527242568, 8.949654080970450862913174794835, 9.706703915882371109975214963135
