| L(s) = 1 | + 1.41i·3-s + (−1.73 − 1.41i)5-s − 2.44i·7-s + 0.999·9-s − 3.46·11-s + (2.00 − 2.44i)15-s + 4.89i·17-s − 3.46·19-s + 3.46·21-s + 2.44i·23-s + (0.999 + 4.89i)25-s + 5.65i·27-s − 4·31-s − 4.89i·33-s + (−3.46 + 4.24i)35-s + ⋯ |
| L(s) = 1 | + 0.816i·3-s + (−0.774 − 0.632i)5-s − 0.925i·7-s + 0.333·9-s − 1.04·11-s + (0.516 − 0.632i)15-s + 1.18i·17-s − 0.794·19-s + 0.755·21-s + 0.510i·23-s + (0.199 + 0.979i)25-s + 1.08i·27-s − 0.718·31-s − 0.852i·33-s + (−0.585 + 0.717i)35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.774 - 0.632i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5834340570\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5834340570\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (1.73 + 1.41i)T \) |
| good | 3 | \( 1 - 1.41iT - 3T^{2} \) |
| 7 | \( 1 + 2.44iT - 7T^{2} \) |
| 11 | \( 1 + 3.46T + 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 2.44iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 4T + 31T^{2} \) |
| 37 | \( 1 - 8.48iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 4.24iT - 43T^{2} \) |
| 47 | \( 1 - 7.34iT - 47T^{2} \) |
| 53 | \( 1 + 5.65iT - 53T^{2} \) |
| 59 | \( 1 + 10.3T + 59T^{2} \) |
| 61 | \( 1 - 3.46T + 61T^{2} \) |
| 67 | \( 1 - 4.24iT - 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 4.89iT - 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 9.89iT - 83T^{2} \) |
| 89 | \( 1 + 6T + 89T^{2} \) |
| 97 | \( 1 + 4.89iT - 97T^{2} \) |
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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
−10.10491620448440057908787758854, −9.246392110833803710769967285320, −8.271508898796163038518742358038, −7.73662255630860953095452608722, −6.82665055183887916701429499245, −5.58857852288369279768175229622, −4.60789976175886110819117401140, −4.13426618481679941372016865648, −3.23595619580319017217388366495, −1.45248089271916545466662774137,
0.24485442215289718714250596262, 2.14070290455971065894524215208, 2.82842075271427681871249037610, 4.12223380035957473620802547356, 5.19773158578469773761587705257, 6.15695746623000269353441404058, 7.10053735774386719588208503615, 7.55202373186791367538401262126, 8.393550471190774484122460090881, 9.205012613917113569023940274971
