| L(s) = 1 | + (1.99 − 0.177i)2-s + 1.73·3-s + (3.93 − 0.707i)4-s + (3.45 − 0.307i)6-s + 1.19·7-s + (7.71 − 2.10i)8-s + 2.99·9-s − 8.22i·11-s + (6.81 − 1.22i)12-s − 11.1i·13-s + (2.38 − 0.212i)14-s + (14.9 − 5.57i)16-s + 20.9i·17-s + (5.97 − 0.533i)18-s + 27.9i·19-s + ⋯ |
| L(s) = 1 | + (0.996 − 0.0888i)2-s + 0.577·3-s + (0.984 − 0.176i)4-s + (0.575 − 0.0512i)6-s + 0.170·7-s + (0.964 − 0.263i)8-s + 0.333·9-s − 0.747i·11-s + (0.568 − 0.102i)12-s − 0.860i·13-s + (0.170 − 0.0151i)14-s + (0.937 − 0.348i)16-s + 1.23i·17-s + (0.332 − 0.0296i)18-s + 1.47i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.959 + 0.281i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(3.67330 - 0.528391i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.67330 - 0.528391i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.99 + 0.177i)T \) |
| 3 | \( 1 - 1.73T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 1.19T + 49T^{2} \) |
| 11 | \( 1 + 8.22iT - 121T^{2} \) |
| 13 | \( 1 + 11.1iT - 169T^{2} \) |
| 17 | \( 1 - 20.9iT - 289T^{2} \) |
| 19 | \( 1 - 27.9iT - 361T^{2} \) |
| 23 | \( 1 - 9.48T + 529T^{2} \) |
| 29 | \( 1 + 40.4T + 841T^{2} \) |
| 31 | \( 1 + 55.3iT - 961T^{2} \) |
| 37 | \( 1 - 50.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 73.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 19.0T + 1.84e3T^{2} \) |
| 47 | \( 1 - 18.0T + 2.20e3T^{2} \) |
| 53 | \( 1 - 57.2iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 60.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 21.3T + 3.72e3T^{2} \) |
| 67 | \( 1 - 9.68T + 4.48e3T^{2} \) |
| 71 | \( 1 - 68.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 84.7iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 23.2iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 93.2T + 6.88e3T^{2} \) |
| 89 | \( 1 + 62.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 91.3iT - 9.40e3T^{2} \) |
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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
−11.58432717140352504229463874194, −10.67538762642786597500739938544, −9.837138228253990890233241945781, −8.319146095446808717534579992631, −7.70196127533400712806547847646, −6.27809349551215676432842387059, −5.47819621491180948888572691507, −4.02783062180757913498251550609, −3.18041127283502089390122980813, −1.67952909197211848373730538296,
1.92258740815355644981542689314, 3.12983501676273750247927513841, 4.44470129739072604043624512078, 5.24693942174148903580707030415, 6.94092065662553611805604729006, 7.22213146785473262554284944834, 8.720560707774949075421619615760, 9.639779882505413127660428538595, 10.91720603303885375779997323881, 11.68720540841168176343920221852
