| L(s) = 1 | − i·3-s + (−1.37 + 1.76i)5-s + (0.501 + 0.501i)7-s − 9-s + (2.02 − 2.02i)11-s + (−4.80 − 4.80i)13-s + (1.76 + 1.37i)15-s + 3.48·17-s + (1.94 + 1.94i)19-s + (0.501 − 0.501i)21-s + (−1.96 + 1.96i)23-s + (−1.21 − 4.84i)25-s + i·27-s + (−1.42 + 5.19i)29-s + (−3.14 + 3.14i)31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.615 + 0.788i)5-s + (0.189 + 0.189i)7-s − 0.333·9-s + (0.609 − 0.609i)11-s + (−1.33 − 1.33i)13-s + (0.455 + 0.355i)15-s + 0.844·17-s + (0.446 + 0.446i)19-s + (0.109 − 0.109i)21-s + (−0.410 + 0.410i)23-s + (−0.243 − 0.969i)25-s + 0.192i·27-s + (−0.264 + 0.964i)29-s + (−0.565 + 0.565i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1740 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.537 + 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8728186124\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8728186124\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (1.37 - 1.76i)T \) |
| 29 | \( 1 + (1.42 - 5.19i)T \) |
| good | 7 | \( 1 + (-0.501 - 0.501i)T + 7iT^{2} \) |
| 11 | \( 1 + (-2.02 + 2.02i)T - 11iT^{2} \) |
| 13 | \( 1 + (4.80 + 4.80i)T + 13iT^{2} \) |
| 17 | \( 1 - 3.48T + 17T^{2} \) |
| 19 | \( 1 + (-1.94 - 1.94i)T + 19iT^{2} \) |
| 23 | \( 1 + (1.96 - 1.96i)T - 23iT^{2} \) |
| 31 | \( 1 + (3.14 - 3.14i)T - 31iT^{2} \) |
| 37 | \( 1 - 1.37iT - 37T^{2} \) |
| 41 | \( 1 + (6.67 + 6.67i)T + 41iT^{2} \) |
| 43 | \( 1 + 6.60iT - 43T^{2} \) |
| 47 | \( 1 + 12.7iT - 47T^{2} \) |
| 53 | \( 1 + (-8.12 + 8.12i)T - 53iT^{2} \) |
| 59 | \( 1 + 6.18iT - 59T^{2} \) |
| 61 | \( 1 + (-1.97 + 1.97i)T - 61iT^{2} \) |
| 67 | \( 1 + (7.83 - 7.83i)T - 67iT^{2} \) |
| 71 | \( 1 - 1.63iT - 71T^{2} \) |
| 73 | \( 1 + 11.5T + 73T^{2} \) |
| 79 | \( 1 + (8.73 + 8.73i)T + 79iT^{2} \) |
| 83 | \( 1 + (-8.37 + 8.37i)T - 83iT^{2} \) |
| 89 | \( 1 + (-1.03 - 1.03i)T + 89iT^{2} \) |
| 97 | \( 1 + 15.3iT - 97T^{2} \) |
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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
−8.801441497920611858402596906360, −8.171879313675673342123448797512, −7.29658762081624561224262152120, −6.99730095516569487059638249108, −5.70135691367023201634477219858, −5.22826658497786710487482839335, −3.63307113708635743329273103138, −3.15388553752349827588603884716, −1.90856658254874600093217667525, −0.33716982328306868030258121754,
1.37551790306516408527031064693, 2.71898998331213828020555788873, 4.15053928975712455272458140980, 4.40136793481116512867201042448, 5.27309296895481874767599254054, 6.38174190630957719712488584940, 7.45898180918338143272079357836, 7.85774275683496428198132884019, 9.131021704454102020040430592521, 9.384237403469872525408653341652
