| L(s) = 1 | − 1.73i·3-s + 1.73i·7-s − 2.99·9-s + 5·13-s − 8.66i·19-s + 2.99·21-s + 5.19i·27-s − 8.66i·31-s + 10·37-s − 8.66i·39-s − 12.1i·43-s + 4·49-s − 15·57-s − 13·61-s − 5.19i·63-s + ⋯ |
| L(s) = 1 | − 0.999i·3-s + 0.654i·7-s − 0.999·9-s + 1.38·13-s − 1.98i·19-s + 0.654·21-s + 0.999i·27-s − 1.55i·31-s + 1.64·37-s − 1.38i·39-s − 1.84i·43-s + 0.571·49-s − 1.98·57-s − 1.66·61-s − 0.654i·63-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.539797807\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.539797807\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 1.73iT \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.73iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 13 | \( 1 - 5T + 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 8.66iT - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 + 8.66iT - 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 12.1iT - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 13T + 61T^{2} \) |
| 67 | \( 1 - 15.5iT - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 10T + 73T^{2} \) |
| 79 | \( 1 + 17.3iT - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 5T + 97T^{2} \) |
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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
−9.110824401300649064494489415528, −8.850439930202716412889193890995, −7.86426106080415528937829408504, −7.06933300342397849113927914571, −6.16909240897454492883685038915, −5.61799415787528107734284135978, −4.35004484194610625156092304974, −3.02073701919884995268063813039, −2.13886360736245894790926960656, −0.73445572320676177890159626893,
1.34931575640980451940120621710, 3.11337740610490125154215786245, 3.85588443782433436363421834666, 4.63063445303689785949919255565, 5.80447354023165349599257911722, 6.35857194223539498169326806148, 7.73886512552691845048061528922, 8.351007169137469552578276505397, 9.235249485033766911680955272731, 10.04192331602531944883277762109
