| L(s) = 1 | + (1.5 − 0.866i)3-s − 1.73i·5-s − 1.73i·7-s + (1.5 − 2.59i)9-s − 13-s + (−1.49 − 2.59i)15-s − 1.73i·17-s + 3.46i·19-s + (−1.49 − 2.59i)21-s − 6·23-s + 2.00·25-s − 5.19i·27-s − 6.92i·29-s + 3.46i·31-s − 2.99·35-s + ⋯ |
| L(s) = 1 | + (0.866 − 0.499i)3-s − 0.774i·5-s − 0.654i·7-s + (0.5 − 0.866i)9-s − 0.277·13-s + (−0.387 − 0.670i)15-s − 0.420i·17-s + 0.794i·19-s + (−0.327 − 0.566i)21-s − 1.25·23-s + 0.400·25-s − 0.999i·27-s − 1.28i·29-s + 0.622i·31-s − 0.507·35-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 624 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.33541 - 1.33541i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.33541 - 1.33541i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.5 + 0.866i)T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 + 1.73iT - 5T^{2} \) |
| 7 | \( 1 + 1.73iT - 7T^{2} \) |
| 11 | \( 1 + 11T^{2} \) |
| 17 | \( 1 + 1.73iT - 17T^{2} \) |
| 19 | \( 1 - 3.46iT - 19T^{2} \) |
| 23 | \( 1 + 6T + 23T^{2} \) |
| 29 | \( 1 + 6.92iT - 29T^{2} \) |
| 31 | \( 1 - 3.46iT - 31T^{2} \) |
| 37 | \( 1 - 11T + 37T^{2} \) |
| 41 | \( 1 - 3.46iT - 41T^{2} \) |
| 43 | \( 1 - 1.73iT - 43T^{2} \) |
| 47 | \( 1 + 9T + 47T^{2} \) |
| 53 | \( 1 + 3.46iT - 53T^{2} \) |
| 59 | \( 1 - 12T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 - 10.3iT - 67T^{2} \) |
| 71 | \( 1 - 3T + 71T^{2} \) |
| 73 | \( 1 - 4T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 - 10.3iT - 89T^{2} \) |
| 97 | \( 1 + 8T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.00906674623971017625633080708, −9.611257179216928299427942150151, −8.416981090273499866063672925573, −7.952696707704436140881909050240, −6.99740668979875648041635629143, −5.96536241018065836418971978243, −4.60026047493341262902301323999, −3.73715299752294137116318452325, −2.36047309725842445285004482738, −0.989461695820053346797347378725,
2.15445540848915262839798435037, 3.01613590924030657537488865432, 4.11209571134030514338224508901, 5.24410437913649204457744327043, 6.41968924635963799481812968729, 7.41593799681028271054092943188, 8.291788091446535043741587431832, 9.121816764538392036083944497359, 9.903341818653298487399067790120, 10.69698940322988457701248614866
