L(s) = 1 | + (0.723 − 0.690i)2-s + (−0.142 + 0.989i)3-s + (0.0475 − 0.998i)4-s + (0.415 + 0.909i)5-s + (0.580 + 0.814i)6-s + (0.235 + 0.971i)7-s + (−0.654 − 0.755i)8-s + (−0.959 − 0.281i)9-s + (0.928 + 0.371i)10-s + (0.580 − 0.814i)11-s + (0.981 + 0.189i)12-s + (−0.327 − 0.945i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (−0.995 − 0.0950i)16-s + (0.0475 + 0.998i)17-s + ⋯ |
L(s) = 1 | + (0.723 − 0.690i)2-s + (−0.142 + 0.989i)3-s + (0.0475 − 0.998i)4-s + (0.415 + 0.909i)5-s + (0.580 + 0.814i)6-s + (0.235 + 0.971i)7-s + (−0.654 − 0.755i)8-s + (−0.959 − 0.281i)9-s + (0.928 + 0.371i)10-s + (0.580 − 0.814i)11-s + (0.981 + 0.189i)12-s + (−0.327 − 0.945i)13-s + (0.841 + 0.540i)14-s + (−0.959 + 0.281i)15-s + (−0.995 − 0.0950i)16-s + (0.0475 + 0.998i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 67 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.994 + 0.107i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.285710053 + 0.06947727385i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.285710053 + 0.06947727385i\) |
\(L(1)\) |
\(\approx\) |
\(1.375658067 + 0.002938546210i\) |
\(L(1)\) |
\(\approx\) |
\(1.375658067 + 0.002938546210i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 67 | \( 1 \) |
good | 2 | \( 1 + (0.723 - 0.690i)T \) |
| 3 | \( 1 + (-0.142 + 0.989i)T \) |
| 5 | \( 1 + (0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.235 + 0.971i)T \) |
| 11 | \( 1 + (0.580 - 0.814i)T \) |
| 13 | \( 1 + (-0.327 - 0.945i)T \) |
| 17 | \( 1 + (0.0475 + 0.998i)T \) |
| 19 | \( 1 + (0.235 - 0.971i)T \) |
| 23 | \( 1 + (-0.786 - 0.618i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 31 | \( 1 + (-0.327 + 0.945i)T \) |
| 37 | \( 1 + (-0.5 + 0.866i)T \) |
| 41 | \( 1 + (-0.888 - 0.458i)T \) |
| 43 | \( 1 + (0.841 - 0.540i)T \) |
| 47 | \( 1 + (0.928 - 0.371i)T \) |
| 53 | \( 1 + (0.841 + 0.540i)T \) |
| 59 | \( 1 + (-0.654 - 0.755i)T \) |
| 61 | \( 1 + (0.580 + 0.814i)T \) |
| 71 | \( 1 + (0.0475 - 0.998i)T \) |
| 73 | \( 1 + (0.580 + 0.814i)T \) |
| 79 | \( 1 + (0.981 + 0.189i)T \) |
| 83 | \( 1 + (-0.995 - 0.0950i)T \) |
| 89 | \( 1 + (-0.142 - 0.989i)T \) |
| 97 | \( 1 + (-0.5 + 0.866i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−31.88930111858292451180171429417, −31.06887451430871737057589835080, −29.84611332323244177720652831510, −29.189959861448969549604262307775, −27.60519661585975021004266566455, −26.051851103688336539582390744231, −25.05834338077038012901880372195, −24.22519542111154857006103410394, −23.435220218633079039174739353601, −22.36146580562618024998569969501, −20.78934380468195502400504097745, −19.95187194699357613948516445942, −18.0660900446695629985550148424, −17.06761135433588216867855147154, −16.3960137394440873254397968796, −14.3607638379334839641883848772, −13.717139261910412673472914259364, −12.54553074261932726160697327457, −11.661275253112375854393950111249, −9.33559226027715656337045454270, −7.78668819441890235290321980588, −6.89099922970307322569058551936, −5.47021131668229920789699471945, −4.14327889509968379862388899508, −1.8067559837071901945037770574,
2.50285139473869148785406473973, 3.64055104245760379597111814691, 5.33550454595489970311731070427, 6.21319983321410020918747539369, 8.828369446408544747540899103, 10.14073460213497345369966012665, 11.01641860941125686249951535107, 12.110245298622091246696670053261, 13.825301177860578348610189437825, 14.88003272075852225489714449811, 15.55651830152480555004284668864, 17.43071172667464972124071032639, 18.74005212351601924176457312697, 19.93855120098455713668697556236, 21.30293832842658095669985982854, 22.03269585293754624945037368432, 22.50095820619385464293743327732, 24.12253363410172873141452255439, 25.43569097543767091577451362037, 26.816076258413186182820580562125, 27.844172329187596652780065997498, 28.75793542242878592491420414392, 29.99494865889214047182567753131, 30.88039817380503833967838296152, 32.26665587825237868755734915318