L(s) = 1 | − i·2-s + (0.707 − 0.707i)3-s − 4-s + (−0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + i·8-s − i·9-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + 13-s + (−0.707 + 0.707i)14-s + 16-s − 18-s + i·19-s − 21-s + (−0.707 + 0.707i)22-s + ⋯ |
L(s) = 1 | − i·2-s + (0.707 − 0.707i)3-s − 4-s + (−0.707 − 0.707i)6-s + (−0.707 − 0.707i)7-s + i·8-s − i·9-s + (−0.707 − 0.707i)11-s + (−0.707 + 0.707i)12-s + 13-s + (−0.707 + 0.707i)14-s + 16-s − 18-s + i·19-s − 21-s + (−0.707 + 0.707i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.758 - 0.651i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3478100082 - 0.9390469205i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3478100082 - 0.9390469205i\) |
\(L(1)\) |
\(\approx\) |
\(0.7247234035 - 0.7805983208i\) |
\(L(1)\) |
\(\approx\) |
\(0.7247234035 - 0.7805983208i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 17 | \( 1 \) |
good | 2 | \( 1 - iT \) |
| 3 | \( 1 + (0.707 - 0.707i)T \) |
| 7 | \( 1 + (-0.707 - 0.707i)T \) |
| 11 | \( 1 + (-0.707 - 0.707i)T \) |
| 13 | \( 1 + T \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + (0.707 + 0.707i)T \) |
| 29 | \( 1 + (0.707 - 0.707i)T \) |
| 31 | \( 1 + (-0.707 + 0.707i)T \) |
| 37 | \( 1 + (0.707 - 0.707i)T \) |
| 41 | \( 1 + (0.707 + 0.707i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 - iT \) |
| 61 | \( 1 + (0.707 + 0.707i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.707 + 0.707i)T \) |
| 73 | \( 1 + (-0.707 + 0.707i)T \) |
| 79 | \( 1 + (-0.707 - 0.707i)T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.707 + 0.707i)T \) |
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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
−31.30456446563388292500143978901, −30.76070546012272868482751214682, −28.599034147741302117630571229023, −27.840848339610381907698202117756, −26.605781005458273664641505627924, −25.69517564849393624004259681484, −25.27012312206083330041252889268, −23.80343196942068078837831826185, −22.61664495636251718846292089532, −21.73367146430478683024153394307, −20.49265375517073911169627846266, −19.09160126422064896851795081173, −18.13047037985524952869834069843, −16.59034032318196149491474047020, −15.62738890902798858471001769511, −15.07424001522578746235243674691, −13.674006835809029951478155382576, −12.74156991647335659659540656092, −10.56552429175653442477372506210, −9.31055153271395306910335205710, −8.55818630834904285081944796824, −7.13669291021869687880779248388, −5.63105648180611671401780699636, −4.36653930811977800866970174663, −2.85299293783222917403203356982,
1.15832363681212858763789260095, 2.897568113447931841807196569285, 3.85636013568059355256340769581, 5.98000514122303056276591833240, 7.697982717932381577176910422939, 8.81970526940337293071379741961, 10.06883415304507147641414773101, 11.29966036407139295891687605980, 12.80294391030712362857679594749, 13.39734352081556036573853212695, 14.38470149494576533213962054849, 16.15390594983896790226582134186, 17.73310425563286755887464957169, 18.768593283216669130822520053480, 19.49989043946104381519144907248, 20.555301046161030601588965866672, 21.380101606176422853421870699839, 23.07959510594324804485289633813, 23.56439350028804470796122679442, 25.21479007283342120040149316056, 26.289596487660277066939515421519, 27.062381473753784812984000438482, 28.68682765339910678957518402838, 29.39509310169605385573837598812, 30.23500195191550261560026617795