L(s) = 1 | + (0.739 + 0.673i)2-s + (−0.602 + 0.798i)3-s + (0.0922 + 0.995i)4-s + (−0.602 + 0.798i)5-s + (−0.982 + 0.183i)6-s + (−0.982 + 0.183i)7-s + (−0.602 + 0.798i)8-s + (−0.273 − 0.961i)9-s + (−0.982 + 0.183i)10-s + (−0.273 + 0.961i)11-s + (−0.850 − 0.526i)12-s + (0.0922 − 0.995i)13-s + (−0.850 − 0.526i)14-s + (−0.273 − 0.961i)15-s + (−0.982 + 0.183i)16-s + (0.445 − 0.895i)17-s + ⋯ |
L(s) = 1 | + (0.739 + 0.673i)2-s + (−0.602 + 0.798i)3-s + (0.0922 + 0.995i)4-s + (−0.602 + 0.798i)5-s + (−0.982 + 0.183i)6-s + (−0.982 + 0.183i)7-s + (−0.602 + 0.798i)8-s + (−0.273 − 0.961i)9-s + (−0.982 + 0.183i)10-s + (−0.273 + 0.961i)11-s + (−0.850 − 0.526i)12-s + (0.0922 − 0.995i)13-s + (−0.850 − 0.526i)14-s + (−0.273 − 0.961i)15-s + (−0.982 + 0.183i)16-s + (0.445 − 0.895i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 919 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.837 - 0.546i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.001601027671 + 0.0004765820772i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.001601027671 + 0.0004765820772i\) |
\(L(1)\) |
\(\approx\) |
\(0.5205594258 + 0.5800632127i\) |
\(L(1)\) |
\(\approx\) |
\(0.5205594258 + 0.5800632127i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 919 | \( 1 \) |
good | 2 | \( 1 + (0.739 + 0.673i)T \) |
| 3 | \( 1 + (-0.602 + 0.798i)T \) |
| 5 | \( 1 + (-0.602 + 0.798i)T \) |
| 7 | \( 1 + (-0.982 + 0.183i)T \) |
| 11 | \( 1 + (-0.273 + 0.961i)T \) |
| 13 | \( 1 + (0.0922 - 0.995i)T \) |
| 17 | \( 1 + (0.445 - 0.895i)T \) |
| 19 | \( 1 + (-0.602 - 0.798i)T \) |
| 23 | \( 1 + (-0.273 + 0.961i)T \) |
| 29 | \( 1 + (0.739 + 0.673i)T \) |
| 31 | \( 1 + (-0.982 + 0.183i)T \) |
| 37 | \( 1 + (0.0922 - 0.995i)T \) |
| 41 | \( 1 + (-0.850 + 0.526i)T \) |
| 43 | \( 1 + (-0.602 - 0.798i)T \) |
| 47 | \( 1 + (0.0922 + 0.995i)T \) |
| 53 | \( 1 + T \) |
| 59 | \( 1 + (0.0922 + 0.995i)T \) |
| 61 | \( 1 + (-0.982 - 0.183i)T \) |
| 67 | \( 1 + (0.932 + 0.361i)T \) |
| 71 | \( 1 + (-0.982 + 0.183i)T \) |
| 73 | \( 1 + (0.932 + 0.361i)T \) |
| 79 | \( 1 + (-0.982 - 0.183i)T \) |
| 83 | \( 1 + (0.445 + 0.895i)T \) |
| 89 | \( 1 + (-0.602 - 0.798i)T \) |
| 97 | \( 1 + (-0.273 - 0.961i)T \) |
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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
−21.92583271560522737319810624960, −21.336717794600197135808364555571, −20.28129973405728460463515147068, −19.53420364720486295748481344716, −18.91846926227878070478137045590, −18.53849414003282834913150397808, −16.80192172103177718906630163632, −16.60148274872308377930342678733, −15.68914125393777310178976578325, −14.52111413090835532251848788541, −13.53039809327209270318396952236, −13.06600505449355795229703835399, −12.23769228036502127251278513235, −11.82113829526230421679165761988, −10.811868073955387788740160133631, −10.0851823162618700654571422876, −8.82743811607081788065791815773, −8.01660482090792375858759309872, −6.65872844939870324868587075227, −6.159328676021600704643396479590, −5.24737603998244011223657483752, −4.18960730556553685240409439414, −3.434779706261632924688351166718, −2.141173519510793663393027380945, −1.11888205274887335295247546251,
0.00067014020375354999814887573, 2.7026888251766093040058664360, 3.31309515401994228173520394627, 4.16647317319740774966513044566, 5.155971812087595668982119088628, 5.880770007190537580076590443, 6.919473717502583591620954483880, 7.35128569329365873741390810523, 8.659101958936763223406496755879, 9.6947044093681224833286485405, 10.50695219343791535482560615551, 11.4151808047898705384384312919, 12.25623351403226142976935003702, 12.85282093786553708015268762985, 13.996798058233073312745378304412, 15.01260310727296203862589126567, 15.43252498302058426195414112711, 15.96160004779234394352956638399, 16.77556686979202425416161533244, 17.83313856803135405895199218131, 18.24101483400099762417567452403, 19.73191354511710620406394949770, 20.29845653431048326479926592985, 21.423586534612766643946315761607, 22.08193981705864448575819198968