| L(s) = 1 | + 2.90·3-s + 3.52·7-s + 5.42·9-s + 3.80·11-s − 2.62·13-s + 5.80·17-s − 5.05·19-s + 10.2·21-s + 0.474·23-s + 7.05·27-s + 2·29-s − 2.75·31-s + 11.0·33-s − 7.18·37-s − 7.61·39-s + 5.18·41-s − 1.95·43-s − 5.33·47-s + 5.42·49-s + 16.8·51-s − 5.37·53-s − 14.6·57-s + 5.05·59-s + 12.2·61-s + 19.1·63-s − 7.76·67-s + 1.37·69-s + ⋯ |
| L(s) = 1 | + 1.67·3-s + 1.33·7-s + 1.80·9-s + 1.14·11-s − 0.727·13-s + 1.40·17-s − 1.15·19-s + 2.23·21-s + 0.0989·23-s + 1.35·27-s + 0.371·29-s − 0.494·31-s + 1.92·33-s − 1.18·37-s − 1.21·39-s + 0.809·41-s − 0.297·43-s − 0.777·47-s + 0.775·49-s + 2.36·51-s − 0.738·53-s − 1.94·57-s + 0.657·59-s + 1.56·61-s + 2.41·63-s − 0.948·67-s + 0.165·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.364128184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.364128184\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 - 2.90T + 3T^{2} \) |
| 7 | \( 1 - 3.52T + 7T^{2} \) |
| 11 | \( 1 - 3.80T + 11T^{2} \) |
| 13 | \( 1 + 2.62T + 13T^{2} \) |
| 17 | \( 1 - 5.80T + 17T^{2} \) |
| 19 | \( 1 + 5.05T + 19T^{2} \) |
| 23 | \( 1 - 0.474T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 2.75T + 31T^{2} \) |
| 37 | \( 1 + 7.18T + 37T^{2} \) |
| 41 | \( 1 - 5.18T + 41T^{2} \) |
| 43 | \( 1 + 1.95T + 43T^{2} \) |
| 47 | \( 1 + 5.33T + 47T^{2} \) |
| 53 | \( 1 + 5.37T + 53T^{2} \) |
| 59 | \( 1 - 5.05T + 59T^{2} \) |
| 61 | \( 1 - 12.2T + 61T^{2} \) |
| 67 | \( 1 + 7.76T + 67T^{2} \) |
| 71 | \( 1 + 4.85T + 71T^{2} \) |
| 73 | \( 1 + 6.66T + 73T^{2} \) |
| 79 | \( 1 + 5.24T + 79T^{2} \) |
| 83 | \( 1 - 12.1T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 13.8T + 97T^{2} \) |
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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
−8.539210373545411819051360781900, −8.097863952734292722905132896592, −7.42144560526949132968217235751, −6.71824801536930619791864554633, −5.48299834837077359173317608808, −4.54998967188077940003743899656, −3.89922648554738463165874093992, −3.03592046566992109231946390370, −2.02364358752162649883944355094, −1.37865335869888555938128545745,
1.37865335869888555938128545745, 2.02364358752162649883944355094, 3.03592046566992109231946390370, 3.89922648554738463165874093992, 4.54998967188077940003743899656, 5.48299834837077359173317608808, 6.71824801536930619791864554633, 7.42144560526949132968217235751, 8.097863952734292722905132896592, 8.539210373545411819051360781900
