L(s) = 1 | + (0.713 + 0.700i)3-s + (−0.905 + 0.424i)5-s + (−0.999 − 0.0250i)7-s + (0.0187 + 0.999i)9-s + (0.900 + 0.435i)11-s + (0.610 − 0.791i)13-s + (−0.943 − 0.331i)15-s + (0.977 + 0.211i)17-s + (−0.988 + 0.149i)19-s + (−0.695 − 0.718i)21-s + (−0.659 − 0.752i)23-s + (0.640 − 0.768i)25-s + (−0.686 + 0.726i)27-s + (0.384 − 0.923i)29-s + (−0.528 + 0.848i)31-s + ⋯ |
L(s) = 1 | + (0.713 + 0.700i)3-s + (−0.905 + 0.424i)5-s + (−0.999 − 0.0250i)7-s + (0.0187 + 0.999i)9-s + (0.900 + 0.435i)11-s + (0.610 − 0.791i)13-s + (−0.943 − 0.331i)15-s + (0.977 + 0.211i)17-s + (−0.988 + 0.149i)19-s + (−0.695 − 0.718i)21-s + (−0.659 − 0.752i)23-s + (0.640 − 0.768i)25-s + (−0.686 + 0.726i)27-s + (0.384 − 0.923i)29-s + (−0.528 + 0.848i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.181 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1631942710 - 0.1960762855i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1631942710 - 0.1960762855i\) |
\(L(1)\) |
\(\approx\) |
\(0.8641762014 + 0.2592752627i\) |
\(L(1)\) |
\(\approx\) |
\(0.8641762014 + 0.2592752627i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 \) |
good | 3 | \( 1 + (0.713 + 0.700i)T \) |
| 5 | \( 1 + (-0.905 + 0.424i)T \) |
| 7 | \( 1 + (-0.999 - 0.0250i)T \) |
| 11 | \( 1 + (0.900 + 0.435i)T \) |
| 13 | \( 1 + (0.610 - 0.791i)T \) |
| 17 | \( 1 + (0.977 + 0.211i)T \) |
| 19 | \( 1 + (-0.988 + 0.149i)T \) |
| 23 | \( 1 + (-0.659 - 0.752i)T \) |
| 29 | \( 1 + (0.384 - 0.923i)T \) |
| 31 | \( 1 + (-0.528 + 0.848i)T \) |
| 37 | \( 1 + (-0.668 - 0.743i)T \) |
| 41 | \( 1 + (-0.747 - 0.663i)T \) |
| 43 | \( 1 + (-0.803 + 0.595i)T \) |
| 47 | \( 1 + (-0.580 + 0.814i)T \) |
| 53 | \( 1 + (0.988 + 0.149i)T \) |
| 59 | \( 1 + (-0.845 - 0.533i)T \) |
| 61 | \( 1 + (-0.337 + 0.941i)T \) |
| 67 | \( 1 + (-0.943 + 0.331i)T \) |
| 71 | \( 1 + (0.974 + 0.223i)T \) |
| 73 | \( 1 + (-0.0312 - 0.999i)T \) |
| 79 | \( 1 + (-0.143 + 0.989i)T \) |
| 83 | \( 1 + (-0.772 + 0.635i)T \) |
| 89 | \( 1 + (0.301 - 0.953i)T \) |
| 97 | \( 1 + (-0.106 - 0.994i)T \) |
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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
−18.78142969285569154023323976035, −18.38223651240859693157159897795, −17.07792685926850447867079634731, −16.63422972947157428686344661333, −15.987143540616163181393228749476, −15.17332946133283444040738665326, −14.62708356880135473441595909434, −13.676627656515058576060003576404, −13.349363884231584290260397105, −12.405627433027123760039502766, −11.97514556980061780733514914381, −11.40280721280407067282920830414, −10.26781642849056266164687219750, −9.39261600793060575694810268208, −8.857673350869942548886526364222, −8.30501139926985740613843709194, −7.50517127490679731493680942016, −6.68286258803826621425854805626, −6.346984899007719067706725385656, −5.24622664379658668877364172165, −3.97628257863316371294297039838, −3.66950755899453403234085937504, −3.00601846795683367526203900195, −1.78123811964524323016740802107, −1.1102467902723555886958936709,
0.06768027595177424360954414018, 1.51566186035096806375035001133, 2.65620626265694919695106681419, 3.30501752305462585438437687784, 3.89494894654653731627877631571, 4.384642230665431283207795626917, 5.56293363358494069265264618815, 6.40724258669968621239156290071, 7.09679658551080775007771403579, 7.99069270861111308245154330392, 8.49951662619053478460432031969, 9.23599641450411545162194205262, 10.22807452961395392813559424073, 10.35729096643640970419037271415, 11.297659958996974239466067090230, 12.31580930021670064090538557950, 12.621745548502449983332615887203, 13.70510937504760763202248332666, 14.30412825903691116019768303457, 15.09415626297915856152402927007, 15.35538991358627405853200560435, 16.32655269130262545901395478383, 16.51379325247053521299264920009, 17.56053649021779467660607668047, 18.55589237445720374692416463508