L(s) = 1 | + (1.17 − 0.780i)2-s − 3.02i·3-s + (0.780 − 1.84i)4-s + (−2.35 − 3.56i)6-s + (1.51 − 2.17i)7-s + (−0.516 − 2.78i)8-s − 6.12·9-s + 4.71i·11-s + (−5.56 − 2.35i)12-s − 2·13-s + (0.0846 − 3.74i)14-s + (−2.78 − 2.87i)16-s + 1.12·17-s + (−7.22 + 4.78i)18-s + 4.71·19-s + ⋯ |
L(s) = 1 | + (0.833 − 0.552i)2-s − 1.74i·3-s + (0.390 − 0.920i)4-s + (−0.962 − 1.45i)6-s + (0.570 − 0.821i)7-s + (−0.182 − 0.983i)8-s − 2.04·9-s + 1.42i·11-s + (−1.60 − 0.680i)12-s − 0.554·13-s + (0.0226 − 0.999i)14-s + (−0.695 − 0.718i)16-s + 0.272·17-s + (−1.70 + 1.12i)18-s + 1.08·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 700 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.119027 - 2.41234i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.119027 - 2.41234i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.17 + 0.780i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.51 + 2.17i)T \) |
good | 3 | \( 1 + 3.02iT - 3T^{2} \) |
| 11 | \( 1 - 4.71iT - 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 1.12T + 17T^{2} \) |
| 19 | \( 1 - 4.71T + 19T^{2} \) |
| 23 | \( 1 - 6.41T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 3.39T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 - 1.12iT - 41T^{2} \) |
| 43 | \( 1 + 0.371T + 43T^{2} \) |
| 47 | \( 1 - 5.08iT - 47T^{2} \) |
| 53 | \( 1 + 2iT - 53T^{2} \) |
| 59 | \( 1 - 2.06T + 59T^{2} \) |
| 61 | \( 1 + 2iT - 61T^{2} \) |
| 67 | \( 1 - 3.76T + 67T^{2} \) |
| 71 | \( 1 + 7.36iT - 71T^{2} \) |
| 73 | \( 1 + 15.3T + 73T^{2} \) |
| 79 | \( 1 + 1.32iT - 79T^{2} \) |
| 83 | \( 1 + 3.02iT - 83T^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 + 1.12T + 97T^{2} \) |
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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
−10.26407523875077856841592661217, −9.295037308139830259335991469121, −7.82580914018327674630625278972, −7.21552427722367648899553181956, −6.71005403816213381547714777979, −5.41695271027997173044005197120, −4.59514349168171057083199468683, −3.08439432726161910993295609419, −1.94586344830004656484747044923, −1.04775472227669861390893775886,
2.82047428148093689076399813832, 3.48789893020403113545215980199, 4.65329876802586717297159516877, 5.36709056054356493631371128185, 5.85712394450760710090589587989, 7.35788127861534772224525594602, 8.566338714456822425135101538640, 8.912390993021434752397873339175, 10.00905405004654238938153513703, 11.18097812503075717915563880067