L(s) = 1 | − i·3-s + (−2.07 − 0.832i)5-s − 0.944i·7-s − 9-s − 5.62·11-s + 4.06i·13-s + (−0.832 + 2.07i)15-s + 2.26i·17-s − 2.66·19-s − 0.944·21-s − 4.72i·23-s + (3.61 + 3.45i)25-s + i·27-s − 3.13·29-s + 3.52·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + (−0.928 − 0.372i)5-s − 0.357i·7-s − 0.333·9-s − 1.69·11-s + 1.12i·13-s + (−0.214 + 0.535i)15-s + 0.548i·17-s − 0.612·19-s − 0.206·21-s − 0.984i·23-s + (0.723 + 0.690i)25-s + 0.192i·27-s − 0.582·29-s + 0.633·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.928 + 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9117489764\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9117489764\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.07 + 0.832i)T \) |
| 67 | \( 1 - iT \) |
good | 7 | \( 1 + 0.944iT - 7T^{2} \) |
| 11 | \( 1 + 5.62T + 11T^{2} \) |
| 13 | \( 1 - 4.06iT - 13T^{2} \) |
| 17 | \( 1 - 2.26iT - 17T^{2} \) |
| 19 | \( 1 + 2.66T + 19T^{2} \) |
| 23 | \( 1 + 4.72iT - 23T^{2} \) |
| 29 | \( 1 + 3.13T + 29T^{2} \) |
| 31 | \( 1 - 3.52T + 31T^{2} \) |
| 37 | \( 1 - 7.10iT - 37T^{2} \) |
| 41 | \( 1 - 2.05T + 41T^{2} \) |
| 43 | \( 1 + 1.23iT - 43T^{2} \) |
| 47 | \( 1 + 10.2iT - 47T^{2} \) |
| 53 | \( 1 - 1.98iT - 53T^{2} \) |
| 59 | \( 1 - 3.82T + 59T^{2} \) |
| 61 | \( 1 + 4.66T + 61T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 3.65iT - 73T^{2} \) |
| 79 | \( 1 + 4.89T + 79T^{2} \) |
| 83 | \( 1 - 0.492iT - 83T^{2} \) |
| 89 | \( 1 - 5.50T + 89T^{2} \) |
| 97 | \( 1 + 8.86iT - 97T^{2} \) |
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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
−8.393272726907281224898253490344, −7.68633913679068000066865522708, −7.05536359544228495008662238139, −6.35581379847845554253673090839, −5.34219574859543379641784635692, −4.58605806134690876503503660716, −3.89609592819603553036106277139, −2.80744868701169630337050745050, −1.91587339672871949520775624177, −0.57517121842081134036571364140,
0.47390205877290420120528439536, 2.41758260061344086738769362465, 3.01152574852177701260157542208, 3.82782355009107892647032568337, 4.79130573659309244938577440664, 5.40462793702230526698179401755, 6.11161774280310904269581564235, 7.34376708439514382572813884444, 7.75859034963903002796454303178, 8.341164579303965886037010887443