| L(s) = 1 | + 2.21i·5-s + i·7-s − 0.614·11-s − 5.56·13-s − 0.585i·17-s − 3.89i·19-s − 0.850·23-s + 0.0963·25-s − 4.74i·29-s + 3.07i·31-s − 2.21·35-s + 2.17·37-s + 6.07i·41-s − 9.21i·43-s − 3.41·47-s + ⋯ |
| L(s) = 1 | + 0.990i·5-s + 0.377i·7-s − 0.185·11-s − 1.54·13-s − 0.142i·17-s − 0.894i·19-s − 0.177·23-s + 0.0192·25-s − 0.880i·29-s + 0.551i·31-s − 0.374·35-s + 0.357·37-s + 0.949i·41-s − 1.40i·43-s − 0.498·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 + 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.132913368\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.132913368\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - iT \) |
| good | 5 | \( 1 - 2.21iT - 5T^{2} \) |
| 11 | \( 1 + 0.614T + 11T^{2} \) |
| 13 | \( 1 + 5.56T + 13T^{2} \) |
| 17 | \( 1 + 0.585iT - 17T^{2} \) |
| 19 | \( 1 + 3.89iT - 19T^{2} \) |
| 23 | \( 1 + 0.850T + 23T^{2} \) |
| 29 | \( 1 + 4.74iT - 29T^{2} \) |
| 31 | \( 1 - 3.07iT - 31T^{2} \) |
| 37 | \( 1 - 2.17T + 37T^{2} \) |
| 41 | \( 1 - 6.07iT - 41T^{2} \) |
| 43 | \( 1 + 9.21iT - 43T^{2} \) |
| 47 | \( 1 + 3.41T + 47T^{2} \) |
| 53 | \( 1 + 4.47iT - 53T^{2} \) |
| 59 | \( 1 - 13.9T + 59T^{2} \) |
| 61 | \( 1 + 8.92T + 61T^{2} \) |
| 67 | \( 1 + 4.83iT - 67T^{2} \) |
| 71 | \( 1 + 5.33T + 71T^{2} \) |
| 73 | \( 1 + 9.28T + 73T^{2} \) |
| 79 | \( 1 - 1.29iT - 79T^{2} \) |
| 83 | \( 1 - 14.5T + 83T^{2} \) |
| 89 | \( 1 + 11.4iT - 89T^{2} \) |
| 97 | \( 1 - 7.27T + 97T^{2} \) |
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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
−7.83589451834374899195925830247, −7.19902256249125588929249701667, −6.72580832654775086262332114759, −5.90694296920907023652283000592, −5.05843716992672568202909606141, −4.46971878115254365059242741488, −3.29995797474232562393786406912, −2.67445716043996174366511224728, −2.01869548569584547670410123845, −0.34256551091459152926923513888,
0.874405994031680972316733739956, 1.88791899265750238759320607617, 2.85499499830057701871566955108, 3.88854853685782810234923855870, 4.63941010776604061777538875572, 5.16794728108004199204999415425, 5.92554432001766297641700165650, 6.83688173657454595450769553474, 7.60449499152718438598184608489, 8.037681291870861842998049058148
