| L(s) = 1 | − 2.55·3-s + (0.790 − 2.09i)5-s + i·7-s + 3.53·9-s − 2.94i·11-s + 1.88·13-s + (−2.02 + 5.34i)15-s + 1.25i·17-s − 2.48i·19-s − 2.55i·21-s + 2.92i·23-s + (−3.74 − 3.30i)25-s − 1.36·27-s + 0.808i·29-s + 10.5·31-s + ⋯ |
| L(s) = 1 | − 1.47·3-s + (0.353 − 0.935i)5-s + 0.377i·7-s + 1.17·9-s − 0.886i·11-s + 0.522·13-s + (−0.521 + 1.38i)15-s + 0.304i·17-s − 0.570i·19-s − 0.557i·21-s + 0.610i·23-s + (−0.749 − 0.661i)25-s − 0.262·27-s + 0.150i·29-s + 1.88·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7136496530\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7136496530\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.790 + 2.09i)T \) |
| 7 | \( 1 - iT \) |
| good | 3 | \( 1 + 2.55T + 3T^{2} \) |
| 11 | \( 1 + 2.94iT - 11T^{2} \) |
| 13 | \( 1 - 1.88T + 13T^{2} \) |
| 17 | \( 1 - 1.25iT - 17T^{2} \) |
| 19 | \( 1 + 2.48iT - 19T^{2} \) |
| 23 | \( 1 - 2.92iT - 23T^{2} \) |
| 29 | \( 1 - 0.808iT - 29T^{2} \) |
| 31 | \( 1 - 10.5T + 31T^{2} \) |
| 37 | \( 1 - 2.08T + 37T^{2} \) |
| 41 | \( 1 + 9.10T + 41T^{2} \) |
| 43 | \( 1 + 11.5T + 43T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 + 9.17T + 53T^{2} \) |
| 59 | \( 1 + 10.0iT - 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 2.89T + 67T^{2} \) |
| 71 | \( 1 + 11.4T + 71T^{2} \) |
| 73 | \( 1 + 16.4iT - 73T^{2} \) |
| 79 | \( 1 - 13.0T + 79T^{2} \) |
| 83 | \( 1 + 8.79T + 83T^{2} \) |
| 89 | \( 1 - 6.64T + 89T^{2} \) |
| 97 | \( 1 + 7.33iT - 97T^{2} \) |
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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
−9.692930806217895474912865850107, −8.678888203202482737974601284361, −8.099322204256430037002645990220, −6.56525663181653444727123006997, −6.17466839921212082478298365032, −5.21670524984095786217339544421, −4.78653665772550061914881545486, −3.38250855498399584187028432892, −1.62089645854786207405495950477, −0.41585021987133961487797000759,
1.36660776350420817809297270186, 2.85543834238790694112265800480, 4.21622079626501648642996339388, 5.04408983591089227171718435859, 6.09686946612850365654616974371, 6.56098508894224147623576823073, 7.30214112401904160360221093009, 8.355308245690260551380329756880, 9.803165391481394387889703998063, 10.18576500255776696601664487579
