L(s) = 1 | + (−0.153 − 0.988i)2-s + (−0.982 − 0.183i)3-s + (−0.952 + 0.303i)4-s + (0.969 + 0.243i)5-s + (−0.0307 + 0.999i)6-s + (0.552 + 0.833i)7-s + (0.445 + 0.895i)8-s + (0.932 + 0.361i)9-s + (0.0922 − 0.995i)10-s + (−0.779 + 0.626i)11-s + (0.992 − 0.122i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (−0.908 − 0.417i)15-s + (0.816 − 0.577i)16-s + (−0.0307 − 0.999i)17-s + ⋯ |
L(s) = 1 | + (−0.153 − 0.988i)2-s + (−0.982 − 0.183i)3-s + (−0.952 + 0.303i)4-s + (0.969 + 0.243i)5-s + (−0.0307 + 0.999i)6-s + (0.552 + 0.833i)7-s + (0.445 + 0.895i)8-s + (0.932 + 0.361i)9-s + (0.0922 − 0.995i)10-s + (−0.779 + 0.626i)11-s + (0.992 − 0.122i)12-s + (0.445 − 0.895i)13-s + (0.739 − 0.673i)14-s + (−0.908 − 0.417i)15-s + (0.816 − 0.577i)16-s + (−0.0307 − 0.999i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 103 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7404312503 - 0.2702666084i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7404312503 - 0.2702666084i\) |
\(L(1)\) |
\(\approx\) |
\(0.7826529369 - 0.2708080028i\) |
\(L(1)\) |
\(\approx\) |
\(0.7826529369 - 0.2708080028i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 103 | \( 1 \) |
good | 2 | \( 1 + (-0.153 - 0.988i)T \) |
| 3 | \( 1 + (-0.982 - 0.183i)T \) |
| 5 | \( 1 + (0.969 + 0.243i)T \) |
| 7 | \( 1 + (0.552 + 0.833i)T \) |
| 11 | \( 1 + (-0.779 + 0.626i)T \) |
| 13 | \( 1 + (0.445 - 0.895i)T \) |
| 17 | \( 1 + (-0.0307 - 0.999i)T \) |
| 19 | \( 1 + (0.650 + 0.759i)T \) |
| 23 | \( 1 + (0.932 - 0.361i)T \) |
| 29 | \( 1 + (-0.696 + 0.717i)T \) |
| 31 | \( 1 + (0.0922 + 0.995i)T \) |
| 37 | \( 1 + (-0.602 - 0.798i)T \) |
| 41 | \( 1 + (0.969 - 0.243i)T \) |
| 43 | \( 1 + (0.992 + 0.122i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.332 - 0.943i)T \) |
| 59 | \( 1 + (0.552 - 0.833i)T \) |
| 61 | \( 1 + (-0.850 + 0.526i)T \) |
| 67 | \( 1 + (-0.998 - 0.0615i)T \) |
| 71 | \( 1 + (-0.696 - 0.717i)T \) |
| 73 | \( 1 + (-0.273 + 0.961i)T \) |
| 79 | \( 1 + (-0.273 - 0.961i)T \) |
| 83 | \( 1 + (-0.998 + 0.0615i)T \) |
| 89 | \( 1 + (0.739 - 0.673i)T \) |
| 97 | \( 1 + (-0.0307 + 0.999i)T \) |
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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
−29.6879678493101268501428453250, −28.682802942406451327363617849198, −27.89092986845926004468584709479, −26.56813761347131612269230023236, −26.06339630021526761925723423847, −24.34314117247748987717266770491, −23.99141161773985035466958809953, −22.90249914747136287503856134731, −21.6951177779384499285612029439, −20.946588458838744947567278018716, −18.92267294082599904856656921543, −17.883639340490033281067475486362, −17.09496518988012439587416784006, −16.453261706086624021647189775204, −15.19445506329606284775789581128, −13.734122996554950408569500370822, −13.11408282032089127240482765283, −11.18030127660626018558548568752, −10.18759844227247031252274379142, −8.99943553177065915172894425927, −7.46777814572795954499274795980, −6.25867232508700236828660478451, −5.33672263728583927164800799692, −4.232781962571556511406257333509, −1.19924211301923770680080413257,
1.478515321856110940758922123630, 2.78776969561075764017360678571, 4.99338125757689895265531658128, 5.619226578155656379674509803447, 7.503449848369767293886473449140, 9.12994260875876941159046403108, 10.304384363237977430573528460550, 11.0856974575146460992948323728, 12.35488898604860467836992279235, 13.086482429606886506390748007437, 14.457018847409149230004962526226, 16.07972200856288381901451595322, 17.67240180987779001651295287087, 17.992375022046009268760666774522, 18.80218608953340130814492089110, 20.69297591574056629468462333050, 21.20867287037538030201890563921, 22.45548401415111403186457463991, 22.921572422427114827715115324132, 24.53265496371592693555594449791, 25.57850132289730352296535175634, 27.04607439865187877824630182601, 27.93800643479284134175670707409, 28.80843082438082989435859134606, 29.42299726671044104086972747812