| L(s) = 1 | + (−2.44 − i)7-s + 4.89·11-s + 4·13-s + 4.89i·17-s + 4i·19-s − 6i·23-s − 5·25-s − 4.89i·29-s + 4.89·31-s + 9.79i·37-s + 4.89i·41-s − 9.79·43-s + 9.79·47-s + (4.99 + 4.89i)49-s − 4.89i·53-s + ⋯ |
| L(s) = 1 | + (−0.925 − 0.377i)7-s + 1.47·11-s + 1.10·13-s + 1.18i·17-s + 0.917i·19-s − 1.25i·23-s − 25-s − 0.909i·29-s + 0.879·31-s + 1.61i·37-s + 0.765i·41-s − 1.49·43-s + 1.42·47-s + (0.714 + 0.699i)49-s − 0.672i·53-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.921 - 0.387i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.868077680\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.868077680\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.44 + i)T \) |
| good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 - 4.89iT - 17T^{2} \) |
| 19 | \( 1 - 4iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 4.89iT - 29T^{2} \) |
| 31 | \( 1 - 4.89T + 31T^{2} \) |
| 37 | \( 1 - 9.79iT - 37T^{2} \) |
| 41 | \( 1 - 4.89iT - 41T^{2} \) |
| 43 | \( 1 + 9.79T + 43T^{2} \) |
| 47 | \( 1 - 9.79T + 47T^{2} \) |
| 53 | \( 1 + 4.89iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 6iT - 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 2iT - 79T^{2} \) |
| 83 | \( 1 - 12iT - 83T^{2} \) |
| 89 | \( 1 - 4.89iT - 89T^{2} \) |
| 97 | \( 1 - 9.79iT - 97T^{2} \) |
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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
−8.385074563164535503559172882749, −7.989784328849599264754449774793, −6.54790735843407778091026017902, −6.51845730503864037010165214654, −5.83897460067769595441525758938, −4.43942810415134553019919364114, −3.85968084541713793246945385763, −3.27114405412193881302624441916, −1.89706008551746634608859974927, −0.930995352016043496874924306019,
0.70151228527838247715945987701, 1.86576026345745395560784307929, 3.07688957069985541041372939222, 3.66515028020494622043670021364, 4.51876266287947387704665993412, 5.66944461993481870820754620317, 6.10017852744262558642035129660, 7.01556377983068838812008955216, 7.39761792081022480905440089064, 8.818920186509554958463570736277
