| L(s) = 1 | − 3.19i·5-s + 1.57·11-s − 0.670·13-s + 5.91i·17-s − 3.28i·19-s − 3.95·23-s − 5.18·25-s + 0.585i·29-s + 9.54i·31-s + 7.69·37-s − 3.77i·41-s − 12.3i·43-s − 7.34·47-s − 13.3i·53-s − 5.02i·55-s + ⋯ |
| L(s) = 1 | − 1.42i·5-s + 0.474·11-s − 0.185·13-s + 1.43i·17-s − 0.754i·19-s − 0.824·23-s − 1.03·25-s + 0.108i·29-s + 1.71i·31-s + 1.26·37-s − 0.589i·41-s − 1.88i·43-s − 1.07·47-s − 1.83i·53-s − 0.677i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6620836226\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6620836226\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 5 | \( 1 + 3.19iT - 5T^{2} \) |
| 11 | \( 1 - 1.57T + 11T^{2} \) |
| 13 | \( 1 + 0.670T + 13T^{2} \) |
| 17 | \( 1 - 5.91iT - 17T^{2} \) |
| 19 | \( 1 + 3.28iT - 19T^{2} \) |
| 23 | \( 1 + 3.95T + 23T^{2} \) |
| 29 | \( 1 - 0.585iT - 29T^{2} \) |
| 31 | \( 1 - 9.54iT - 31T^{2} \) |
| 37 | \( 1 - 7.69T + 37T^{2} \) |
| 41 | \( 1 + 3.77iT - 41T^{2} \) |
| 43 | \( 1 + 12.3iT - 43T^{2} \) |
| 47 | \( 1 + 7.34T + 47T^{2} \) |
| 53 | \( 1 + 13.3iT - 53T^{2} \) |
| 59 | \( 1 + 4.96T + 59T^{2} \) |
| 61 | \( 1 + 6.51T + 61T^{2} \) |
| 67 | \( 1 + 5.51iT - 67T^{2} \) |
| 71 | \( 1 + 6.85T + 71T^{2} \) |
| 73 | \( 1 + 5.84T + 73T^{2} \) |
| 79 | \( 1 + 6.08iT - 79T^{2} \) |
| 83 | \( 1 + 9.48T + 83T^{2} \) |
| 89 | \( 1 + 14.8iT - 89T^{2} \) |
| 97 | \( 1 - 0.183T + 97T^{2} \) |
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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
−7.72192077951890600489040281840, −6.84071862187394267696365898519, −6.12583677819278586222679024013, −5.38126721021395248188245636003, −4.72109218690263535331045154162, −4.09265630719985602647806869463, −3.28864215375067586589256983644, −1.98098193936936145231764000281, −1.32850576918200633700920203462, −0.15909219344036743535934782899,
1.36315496592966898071992735738, 2.61802266422638002944993359833, 2.90480341479209472251305304646, 3.99136514157914607730581427098, 4.56960622856744324102213272913, 5.76780931556662703292716137605, 6.23059870253749892182965055560, 6.86404662202702038236369893897, 7.77369343629911763348643869066, 7.83997844978189795552917001289
