| L(s) = 1 | + 1.50i·2-s + 0.732·3-s − 0.267·4-s − 1.73·5-s + 1.10i·6-s − 2.60i·7-s + 2.60i·8-s − 2.46·9-s − 2.60i·10-s − 1.50i·11-s − 0.196·12-s + 3·13-s + 3.92·14-s − 1.26·15-s − 4.46·16-s − 4.11i·17-s + ⋯ |
| L(s) = 1 | + 1.06i·2-s + 0.422·3-s − 0.133·4-s − 0.774·5-s + 0.450i·6-s − 0.985i·7-s + 0.922i·8-s − 0.821·9-s − 0.824i·10-s − 0.454i·11-s − 0.0566·12-s + 0.832·13-s + 1.04·14-s − 0.327·15-s − 1.11·16-s − 0.997i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.409 - 0.912i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.786349 + 0.509138i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.786349 + 0.509138i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 61 | \( 1 + (3.19 - 7.12i)T \) |
| good | 2 | \( 1 - 1.50iT - 2T^{2} \) |
| 3 | \( 1 - 0.732T + 3T^{2} \) |
| 5 | \( 1 + 1.73T + 5T^{2} \) |
| 7 | \( 1 + 2.60iT - 7T^{2} \) |
| 11 | \( 1 + 1.50iT - 11T^{2} \) |
| 13 | \( 1 - 3T + 13T^{2} \) |
| 17 | \( 1 + 4.11iT - 17T^{2} \) |
| 19 | \( 1 - 1.26T + 19T^{2} \) |
| 23 | \( 1 - 6.72iT - 23T^{2} \) |
| 29 | \( 1 - 4.11iT - 29T^{2} \) |
| 31 | \( 1 - 7.12iT - 31T^{2} \) |
| 37 | \( 1 + 7.12iT - 37T^{2} \) |
| 41 | \( 1 + 6.46T + 41T^{2} \) |
| 43 | \( 1 - 1.90iT - 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 + 11.2iT - 53T^{2} \) |
| 59 | \( 1 + 0.403iT - 59T^{2} \) |
| 67 | \( 1 + 11.6iT - 67T^{2} \) |
| 71 | \( 1 - 1.10iT - 71T^{2} \) |
| 73 | \( 1 - 9.19T + 73T^{2} \) |
| 79 | \( 1 - 14.9iT - 79T^{2} \) |
| 83 | \( 1 - 2.53T + 83T^{2} \) |
| 89 | \( 1 + 13.4iT - 89T^{2} \) |
| 97 | \( 1 - 3.46T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.47335101967697623980621084933, −14.16271660884682132831036397325, −13.69688635012594793162359160796, −11.72094346233528637214005337944, −10.90013841680886612145723590661, −8.962232788882795828857086913424, −7.900081636807882538073969739267, −7.03217454741508072414369878636, −5.46706070633257497969266177984, −3.50630949176599391767065165119,
2.46969311497484853160447964840, 3.92299876522347646487262538340, 6.14924306188065995732180826784, 8.030520026514393478372312204974, 9.084334516971408793327624163756, 10.52715053364718861458157209407, 11.64592572715150661520152084416, 12.24907427920553436159087321970, 13.49770644344522098905453529187, 15.04491694626826444985086730183
