| L(s) = 1 | − 2.45·3-s + 1.45i·5-s − i·7-s + 3.02·9-s − 1.64i·11-s + (−2.45 + 2.64i)13-s − 3.57i·15-s + 5.29·17-s − 0.640i·19-s + 2.45i·21-s − 2.18·23-s + 2.88·25-s − 0.0679·27-s − 2.21·29-s + 4.29i·31-s + ⋯ |
| L(s) = 1 | − 1.41·3-s + 0.650i·5-s − 0.377i·7-s + 1.00·9-s − 0.494i·11-s + (−0.680 + 0.732i)13-s − 0.922i·15-s + 1.28·17-s − 0.146i·19-s + 0.535i·21-s − 0.455·23-s + 0.576·25-s − 0.0130·27-s − 0.410·29-s + 0.771i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1456 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.732 - 0.680i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4294976247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4294976247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + iT \) |
| 13 | \( 1 + (2.45 - 2.64i)T \) |
| good | 3 | \( 1 + 2.45T + 3T^{2} \) |
| 5 | \( 1 - 1.45iT - 5T^{2} \) |
| 11 | \( 1 + 1.64iT - 11T^{2} \) |
| 17 | \( 1 - 5.29T + 17T^{2} \) |
| 19 | \( 1 + 0.640iT - 19T^{2} \) |
| 23 | \( 1 + 2.18T + 23T^{2} \) |
| 29 | \( 1 + 2.21T + 29T^{2} \) |
| 31 | \( 1 - 4.29iT - 31T^{2} \) |
| 37 | \( 1 + 3.26iT - 37T^{2} \) |
| 41 | \( 1 + 0.842iT - 41T^{2} \) |
| 43 | \( 1 + 5.12T + 43T^{2} \) |
| 47 | \( 1 - 0.572iT - 47T^{2} \) |
| 53 | \( 1 + 6.25T + 53T^{2} \) |
| 59 | \( 1 - 10.0iT - 59T^{2} \) |
| 61 | \( 1 + 7.21T + 61T^{2} \) |
| 67 | \( 1 - 3.75iT - 67T^{2} \) |
| 71 | \( 1 - 2.42iT - 71T^{2} \) |
| 73 | \( 1 - 1.48iT - 73T^{2} \) |
| 79 | \( 1 + 5.19T + 79T^{2} \) |
| 83 | \( 1 - 17.3iT - 83T^{2} \) |
| 89 | \( 1 + 4.69iT - 89T^{2} \) |
| 97 | \( 1 - 11.5iT - 97T^{2} \) |
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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
−10.10250471081647526613556023209, −9.218407998416006779974372502469, −8.026887001888521035695543928713, −7.11411383582591586957775121937, −6.58761996978756060688029524282, −5.69956714274428481010232760613, −5.02534433108681720202357240828, −3.98125912601885561056956874017, −2.85205758043858981946082843626, −1.25047532756385152391680861404,
0.23334416572029866709703955825, 1.54161302854326368442467961361, 3.08706711215355447159187213853, 4.48447486980052831665950062448, 5.18477659359768521387682771338, 5.71833975512115984295733206286, 6.57545865783867290299426825063, 7.59149136300110368047987794116, 8.284847066741859960738133480746, 9.479885727473689932293087736654
