| L(s) = 1 | + 2i·3-s + 3.46·5-s + (2 + 1.73i)7-s − 9-s − 3.46·11-s + 3.46·13-s + 6.92i·15-s + 2i·19-s + (−3.46 + 4i)21-s − 3.46i·23-s + 6.99·25-s + 4i·27-s + 6i·29-s − 8·31-s − 6.92i·33-s + ⋯ |
| L(s) = 1 | + 1.15i·3-s + 1.54·5-s + (0.755 + 0.654i)7-s − 0.333·9-s − 1.04·11-s + 0.960·13-s + 1.78i·15-s + 0.458i·19-s + (−0.755 + 0.872i)21-s − 0.722i·23-s + 1.39·25-s + 0.769i·27-s + 1.11i·29-s − 1.43·31-s − 1.20i·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0716 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1792 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0716 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.537093725\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.537093725\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-2 - 1.73i)T \) |
| good | 3 | \( 1 - 2iT - 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 3.46T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 2iT - 19T^{2} \) |
| 23 | \( 1 + 3.46iT - 23T^{2} \) |
| 29 | \( 1 - 6iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 2iT - 37T^{2} \) |
| 41 | \( 1 + 6.92iT - 41T^{2} \) |
| 43 | \( 1 - 10.3T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 + 6iT - 53T^{2} \) |
| 59 | \( 1 + 6iT - 59T^{2} \) |
| 61 | \( 1 - 3.46T + 61T^{2} \) |
| 67 | \( 1 - 3.46T + 67T^{2} \) |
| 71 | \( 1 - 3.46iT - 71T^{2} \) |
| 73 | \( 1 - 6.92iT - 73T^{2} \) |
| 79 | \( 1 + 3.46iT - 79T^{2} \) |
| 83 | \( 1 + 6iT - 83T^{2} \) |
| 89 | \( 1 + 6.92iT - 89T^{2} \) |
| 97 | \( 1 + 13.8iT - 97T^{2} \) |
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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
−9.427441147989221749465342198191, −8.945979307412727425448825149928, −8.215814195159878537687215181709, −7.03166766925932660887166896231, −5.84598974630436377765673290252, −5.46332733702856995997139578999, −4.77484478179335369302540123919, −3.65158592882777264273271355498, −2.50588892202341957149153976129, −1.60488561404812693188391975590,
1.01000571017339276369949623800, 1.83957233770709877289524295112, 2.62292988788677763959578900250, 4.12331770300204152270433532727, 5.31390039354101702332540263653, 5.88167872646949444854727333832, 6.69008259656637490972590677662, 7.53819140957394053979307362786, 8.045217027045244029392061587241, 9.084427067543730269907437332285
