L(s) = 1 | + (0.611 − 0.352i)3-s + (2.56 − 0.647i)7-s + (−1.25 + 2.16i)9-s + (2.25 + 3.89i)11-s + 5.09i·13-s + (−1.73 + i)17-s + (−1.54 + 2.67i)19-s + (1.33 − 1.30i)21-s + (−5.01 − 2.89i)23-s + 3.88i·27-s − 9.50·29-s + (−2.70 − 4.68i)31-s + (2.75 + 1.58i)33-s + (6.14 + 3.54i)37-s + (1.79 + 3.11i)39-s + ⋯ |
L(s) = 1 | + (0.352 − 0.203i)3-s + (0.969 − 0.244i)7-s + (−0.416 + 0.722i)9-s + (0.678 + 1.17i)11-s + 1.41i·13-s + (−0.420 + 0.242i)17-s + (−0.354 + 0.614i)19-s + (0.292 − 0.283i)21-s + (−1.04 − 0.604i)23-s + 0.747i·27-s − 1.76·29-s + (−0.485 − 0.841i)31-s + (0.478 + 0.276i)33-s + (1.00 + 0.582i)37-s + (0.287 + 0.498i)39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.269 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.775485730\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.775485730\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-2.56 + 0.647i)T \) |
good | 3 | \( 1 + (-0.611 + 0.352i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-2.25 - 3.89i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 5.09iT - 13T^{2} \) |
| 17 | \( 1 + (1.73 - i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.54 - 2.67i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.01 + 2.89i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9.50T + 29T^{2} \) |
| 31 | \( 1 + (2.70 + 4.68i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-6.14 - 3.54i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.59T + 41T^{2} \) |
| 43 | \( 1 - 4.70iT - 43T^{2} \) |
| 47 | \( 1 + (-8.74 - 5.04i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-8.58 + 4.95i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4 - 6.92i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.45 + 7.71i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.101 + 0.0585i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-7.08 + 4.09i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.09 - 12.2i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 10.7iT - 83T^{2} \) |
| 89 | \( 1 + (-2.04 + 3.54i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
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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
−9.619319676294501045568644390664, −8.883151338136296806062068908000, −8.062328475238945168202131591942, −7.42518151008143506958123308027, −6.60316203161262776437742516227, −5.55121818496659032580912905563, −4.41517263438809034415163584658, −4.02037185710444930354498807169, −2.13730022242838353656258626546, −1.80759637235794296236142481162,
0.67452886084755764838632022488, 2.21200413626699594261940509596, 3.36924383022845737760964762170, 4.05836760090696094979911324632, 5.47206960108049689681810093204, 5.80918173719655878304904470403, 7.04776710290108947623780751980, 7.973143102500435213499053166431, 8.711833997335578178367362661798, 9.105858620420167260002835412365