L(s) = 1 | + (1.99 + 4.79i)3-s + 16.7i·5-s + 7i·7-s + (−19.0 + 19.1i)9-s + 60.9·11-s + 73.9·13-s + (−80.2 + 33.4i)15-s − 16.3i·17-s + 153. i·19-s + (−33.5 + 13.9i)21-s − 111.·23-s − 154.·25-s + (−129. − 52.8i)27-s − 155. i·29-s − 53.7i·31-s + ⋯ |
L(s) = 1 | + (0.384 + 0.923i)3-s + 1.49i·5-s + 0.377i·7-s + (−0.704 + 0.709i)9-s + 1.66·11-s + 1.57·13-s + (−1.38 + 0.575i)15-s − 0.232i·17-s + 1.85i·19-s + (−0.348 + 0.145i)21-s − 1.01·23-s − 1.23·25-s + (−0.926 − 0.377i)27-s − 0.993i·29-s − 0.311i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.794 - 0.607i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(2.266402557\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.266402557\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.99 - 4.79i)T \) |
| 7 | \( 1 - 7iT \) |
good | 5 | \( 1 - 16.7iT - 125T^{2} \) |
| 11 | \( 1 - 60.9T + 1.33e3T^{2} \) |
| 13 | \( 1 - 73.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 16.3iT - 4.91e3T^{2} \) |
| 19 | \( 1 - 153. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 111.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 155. iT - 2.43e4T^{2} \) |
| 31 | \( 1 + 53.7iT - 2.97e4T^{2} \) |
| 37 | \( 1 + 63.0T + 5.06e4T^{2} \) |
| 41 | \( 1 + 458. iT - 6.89e4T^{2} \) |
| 43 | \( 1 - 95.5iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 70.5T + 1.03e5T^{2} \) |
| 53 | \( 1 + 417. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 328.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 637.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 201. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 394.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 977.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 514. iT - 4.93e5T^{2} \) |
| 83 | \( 1 - 1.13e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.03e3iT - 7.04e5T^{2} \) |
| 97 | \( 1 - 995.T + 9.12e5T^{2} \) |
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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
−11.35612126179114198529077003674, −10.49885124818842804967244769914, −9.770535308129801667695814555120, −8.788026496619630614878594962990, −7.84686078845544482117473690220, −6.42791081489545386502700621395, −5.86542857753592898682690635350, −3.88137316150675312279440203659, −3.56616950082561693358435098891, −1.99515660234469666888313485081,
0.855457724976818171848733420485, 1.54944603882179361040935387290, 3.52686869988838570386759980673, 4.60519474638973932862654355865, 6.05780539195405675836546650503, 6.82734243539667047451405164973, 8.128075309400573227434261243662, 8.885651940293002808427680694202, 9.299404811636360982391886082298, 11.04114967971083189304144967821