| 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
−9.205012613917113569023940274971, −8.393550471190774484122460090881, −7.55202373186791367538401262126, −7.10053735774386719588208503615, −6.15695746623000269353441404058, −5.19773158578469773761587705257, −4.12223380035957473620802547356, −2.82842075271427681871249037610, −2.14070290455971065894524215208, −0.24485442215289718714250596262,
1.45248089271916545466662774137, 3.23595619580319017217388366495, 4.13426618481679941372016865648, 4.60789976175886110819117401140, 5.58857852288369279768175229622, 6.82665055183887916701429499245, 7.73662255630860953095452608722, 8.271508898796163038518742358038, 9.246392110833803710769967285320, 10.10491620448440057908787758854
