| L(s) = 1 | + 1.41i·5-s − i·7-s − 2.82·11-s + 7.07i·17-s − 4i·19-s + 2.82·23-s + 2.99·25-s − 4.24i·29-s − 4i·31-s + 1.41·35-s − 2·37-s + 4.24i·41-s + 12i·43-s + 5.65·47-s − 49-s + ⋯ |
| L(s) = 1 | + 0.632i·5-s − 0.377i·7-s − 0.852·11-s + 1.71i·17-s − 0.917i·19-s + 0.589·23-s + 0.599·25-s − 0.787i·29-s − 0.718i·31-s + 0.239·35-s − 0.328·37-s + 0.662i·41-s + 1.82i·43-s + 0.825·47-s − 0.142·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.019495919\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.019495919\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + iT \) |
| good | 5 | \( 1 - 1.41iT - 5T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 7.07iT - 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 2.82T + 23T^{2} \) |
| 29 | \( 1 + 4.24iT - 29T^{2} \) |
| 31 | \( 1 + 4iT - 31T^{2} \) |
| 37 | \( 1 + 2T + 37T^{2} \) |
| 41 | \( 1 - 4.24iT - 41T^{2} \) |
| 43 | \( 1 - 12iT - 43T^{2} \) |
| 47 | \( 1 - 5.65T + 47T^{2} \) |
| 53 | \( 1 - 7.07iT - 53T^{2} \) |
| 59 | \( 1 + 5.65T + 59T^{2} \) |
| 61 | \( 1 + 14T + 61T^{2} \) |
| 67 | \( 1 - 8iT - 67T^{2} \) |
| 71 | \( 1 + 8.48T + 71T^{2} \) |
| 73 | \( 1 + 4T + 73T^{2} \) |
| 79 | \( 1 + 4iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 9.89iT - 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−8.641297851573223116312932414237, −7.82661465398203978913882456620, −7.36301603660913906521686162137, −6.39410296355891468524371175286, −5.95434690363411652960036707489, −4.84013096377988327918815593375, −4.18902021704373022075374569971, −3.13895289967621847740335817129, −2.49408797515078850416467680897, −1.23549489151800430161525398230,
0.30141424692314440093264404930, 1.56547785639401984845300103010, 2.68144490796705539246245880369, 3.41091536663393842058651286923, 4.64231731786706336101304708871, 5.17765520104301133148675328532, 5.73387633600905383720208049414, 6.90032518844077327278780439707, 7.40596766756280794066460181678, 8.279754950078788874588380188994
