| L(s) = 1 | + i·3-s − 2i·7-s + 2·9-s + 11-s + 4i·13-s + 6i·17-s + 8·19-s + 2·21-s − 3i·23-s + 5i·27-s − 5·31-s + i·33-s − i·37-s − 4·39-s − 10i·43-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 0.755i·7-s + 0.666·9-s + 0.301·11-s + 1.10i·13-s + 1.45i·17-s + 1.83·19-s + 0.436·21-s − 0.625i·23-s + 0.962i·27-s − 0.898·31-s + 0.174i·33-s − 0.164i·37-s − 0.640·39-s − 1.52i·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.159092556\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.159092556\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 - iT - 3T^{2} \) |
| 7 | \( 1 + 2iT - 7T^{2} \) |
| 13 | \( 1 - 4iT - 13T^{2} \) |
| 17 | \( 1 - 6iT - 17T^{2} \) |
| 19 | \( 1 - 8T + 19T^{2} \) |
| 23 | \( 1 + 3iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 + 5T + 31T^{2} \) |
| 37 | \( 1 + iT - 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 10iT - 43T^{2} \) |
| 47 | \( 1 - 47T^{2} \) |
| 53 | \( 1 - 6iT - 53T^{2} \) |
| 59 | \( 1 - 3T + 59T^{2} \) |
| 61 | \( 1 + 4T + 61T^{2} \) |
| 67 | \( 1 - iT - 67T^{2} \) |
| 71 | \( 1 + 15T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 - 2T + 79T^{2} \) |
| 83 | \( 1 - 6iT - 83T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + 7iT - 97T^{2} \) |
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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
−8.695463446717145009632970760488, −7.50700605595504378349956440621, −7.21483224794068178871225338496, −6.34039776536889302371137452106, −5.47587460657565924731547680587, −4.56528802534845176075101829937, −3.96312521004927619865570080613, −3.42208638280408150030517365734, −1.97245136586625596906868444876, −1.09312249414062706774388644984,
0.71487897591103813824125481204, 1.66434073440817626084174201772, 2.80695400218705983522056045701, 3.38934568960171691729558107417, 4.65701264648135938152007293543, 5.36778806196894670622615506095, 5.95582056229150782637958848524, 6.97707025066699430532783075772, 7.49509023004236385675479701520, 7.997827130729703101699967579319
