| L(s) = 1 | − 0.386i·3-s − 0.386i·5-s − 4.17·7-s + 2.85·9-s + 3.78i·11-s + i·13-s − 0.149·15-s − 4.49·17-s + 5.25i·19-s + 1.61i·21-s + 6.11·23-s + 4.85·25-s − 2.26i·27-s + 8.88i·29-s − 4.44·31-s + ⋯ |
| L(s) = 1 | − 0.223i·3-s − 0.172i·5-s − 1.57·7-s + 0.950·9-s + 1.14i·11-s + 0.277i·13-s − 0.0385·15-s − 1.09·17-s + 1.20i·19-s + 0.352i·21-s + 1.27·23-s + 0.970·25-s − 0.435i·27-s + 1.64i·29-s − 0.799·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.258 - 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.849765 + 0.652048i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.849765 + 0.652048i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 - iT \) |
| good | 3 | \( 1 + 0.386iT - 3T^{2} \) |
| 5 | \( 1 + 0.386iT - 5T^{2} \) |
| 7 | \( 1 + 4.17T + 7T^{2} \) |
| 11 | \( 1 - 3.78iT - 11T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 - 5.25iT - 19T^{2} \) |
| 23 | \( 1 - 6.11T + 23T^{2} \) |
| 29 | \( 1 - 8.88iT - 29T^{2} \) |
| 31 | \( 1 + 4.44T + 31T^{2} \) |
| 37 | \( 1 + 3.96iT - 37T^{2} \) |
| 41 | \( 1 - 8.11T + 41T^{2} \) |
| 43 | \( 1 - 12.7iT - 43T^{2} \) |
| 47 | \( 1 + 7.40T + 47T^{2} \) |
| 53 | \( 1 - 9.04iT - 53T^{2} \) |
| 59 | \( 1 + 4.56iT - 59T^{2} \) |
| 61 | \( 1 + 1.33iT - 61T^{2} \) |
| 67 | \( 1 + 6.67iT - 67T^{2} \) |
| 71 | \( 1 - 4.59T + 71T^{2} \) |
| 73 | \( 1 + 11.6T + 73T^{2} \) |
| 79 | \( 1 + 2.57T + 79T^{2} \) |
| 83 | \( 1 - 2.67iT - 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 - 0.237T + 97T^{2} \) |
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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
−10.29269393698053613782523380976, −9.440260209622370366840657299712, −9.037677689047097597248224664664, −7.57952233213925392250374192047, −6.88919673915440599803310603439, −6.33342926023916175256050279759, −4.95514855556048435883153854840, −4.03791689394992672344046381053, −2.90261644690359493226035287747, −1.49787860481307108737470332225,
0.54342003938527005428787804391, 2.63372850226856005686934255602, 3.48328081694013264622755395792, 4.53596118504317505136007571442, 5.74529765084153115895403821470, 6.72286758241013777078232745883, 7.13305329972315275160019507634, 8.614124038330900451344093166402, 9.222829535843404070104142612195, 10.00726613767402093281533478340
