| L(s) = 1 | − 2i·5-s + (−2.23 − 1.41i)7-s − 1.41·11-s − 4.47·17-s − 6.32·19-s + 3.16i·23-s + 25-s + 3.16·29-s + 5.65i·31-s + (−2.82 + 4.47i)35-s + 4.47i·37-s − 4.47·41-s + 4i·43-s + 8·47-s + (3.00 + 6.32i)49-s + ⋯ |
| L(s) = 1 | − 0.894i·5-s + (−0.845 − 0.534i)7-s − 0.426·11-s − 1.08·17-s − 1.45·19-s + 0.659i·23-s + 0.200·25-s + 0.587·29-s + 1.01i·31-s + (−0.478 + 0.755i)35-s + 0.735i·37-s − 0.698·41-s + 0.609i·43-s + 1.16·47-s + (0.428 + 0.903i)49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8630541596\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8630541596\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (2.23 + 1.41i)T \) |
| good | 5 | \( 1 + 2iT - 5T^{2} \) |
| 11 | \( 1 + 1.41T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 4.47T + 17T^{2} \) |
| 19 | \( 1 + 6.32T + 19T^{2} \) |
| 23 | \( 1 - 3.16iT - 23T^{2} \) |
| 29 | \( 1 - 3.16T + 29T^{2} \) |
| 31 | \( 1 - 5.65iT - 31T^{2} \) |
| 37 | \( 1 - 4.47iT - 37T^{2} \) |
| 41 | \( 1 + 4.47T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 9.48T + 53T^{2} \) |
| 59 | \( 1 + 8.94iT - 59T^{2} \) |
| 61 | \( 1 - 8.48T + 61T^{2} \) |
| 67 | \( 1 + 2iT - 67T^{2} \) |
| 71 | \( 1 - 9.48iT - 71T^{2} \) |
| 73 | \( 1 + 6.32iT - 73T^{2} \) |
| 79 | \( 1 + 8.94T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 13.4T + 89T^{2} \) |
| 97 | \( 1 - 6.32iT - 97T^{2} \) |
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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
−8.698589596657611160233013478966, −7.915229895454525837441622545347, −6.89174953543147002352387341481, −6.51235738817573842455320507612, −5.51026472071724636204138799257, −4.69591339507623262907819234613, −4.09219156510229214035725080316, −3.10772240877143530047062135313, −2.08289389267028083667379570066, −0.849021045927260880705946263087,
0.30772383878657698685021740017, 2.32779245465469100649060415756, 2.54227279420013564164168531129, 3.70809819280685586578148918932, 4.43870315630731156614259078414, 5.53919972723900729641822013339, 6.29268748026997572955552039801, 6.76548212173817398663595126619, 7.44529037281193270884801594674, 8.623875350489879854849372248640
