| L(s) = 1 | + 13.4i·7-s − 17.6i·11-s + 7.48i·13-s + 16.9·17-s − 10.9·19-s − 21.9·23-s − 47.3i·29-s + 16.9·31-s − 5.53i·37-s − 66.3i·41-s + 38.9i·43-s − 32.5·47-s − 132.·49-s − 11.2·53-s + 31.8i·59-s + ⋯ |
| L(s) = 1 | + 1.92i·7-s − 1.60i·11-s + 0.575i·13-s + 0.998·17-s − 0.577·19-s − 0.952·23-s − 1.63i·29-s + 0.547·31-s − 0.149i·37-s − 1.61i·41-s + 0.906i·43-s − 0.692·47-s − 2.71·49-s − 0.212·53-s + 0.540i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.472 + 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.389392821\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389392821\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 13.4iT - 49T^{2} \) |
| 11 | \( 1 + 17.6iT - 121T^{2} \) |
| 13 | \( 1 - 7.48iT - 169T^{2} \) |
| 17 | \( 1 - 16.9T + 289T^{2} \) |
| 19 | \( 1 + 10.9T + 361T^{2} \) |
| 23 | \( 1 + 21.9T + 529T^{2} \) |
| 29 | \( 1 + 47.3iT - 841T^{2} \) |
| 31 | \( 1 - 16.9T + 961T^{2} \) |
| 37 | \( 1 + 5.53iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 66.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 38.9iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 32.5T + 2.20e3T^{2} \) |
| 53 | \( 1 + 11.2T + 2.80e3T^{2} \) |
| 59 | \( 1 - 31.8iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 46.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 76iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 77.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 94.9iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 6.92T + 6.24e3T^{2} \) |
| 83 | \( 1 + 62.1T + 6.88e3T^{2} \) |
| 89 | \( 1 + 62.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 124. iT - 9.40e3T^{2} \) |
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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
−8.383713863026903886209126811768, −7.77598663822500161893825272039, −6.46483996441856554266808967403, −5.88824174652394140737811575874, −5.56543332697218958133885926086, −4.43681362628719667559394103088, −3.40894990417658219465447380376, −2.63143291552246246274566639318, −1.80636890573119695140440947985, −0.33595367282213386666758391833,
0.948191148567082882339114593744, 1.79200230048540397080831256950, 3.12916673354415999668791424580, 3.95461049263755515477943565001, 4.58643152978600387645766590017, 5.33876968746641801978628591625, 6.60717889073541760019800369465, 6.98402649435342330836534655429, 7.79609389246694275421226874716, 8.175695440376451061418014801397
