L(s) = 1 | + (−0.683 − 1.59i)3-s + (3.40 + 1.96i)5-s + (0.961 − 0.555i)7-s + (−2.06 + 2.17i)9-s + (1.63 + 2.82i)11-s + (0.124 − 0.216i)13-s + (0.802 − 6.77i)15-s − 5.86i·17-s + 2.19i·19-s + (−1.54 − 1.15i)21-s + (2.79 − 4.83i)23-s + (5.24 + 9.09i)25-s + (4.87 + 1.80i)27-s + (2.35 − 1.36i)29-s + (−8.96 − 5.17i)31-s + ⋯ |
L(s) = 1 | + (−0.394 − 0.918i)3-s + (1.52 + 0.880i)5-s + (0.363 − 0.209i)7-s + (−0.688 + 0.725i)9-s + (0.492 + 0.852i)11-s + (0.0346 − 0.0600i)13-s + (0.207 − 1.74i)15-s − 1.42i·17-s + 0.504i·19-s + (−0.336 − 0.251i)21-s + (0.582 − 1.00i)23-s + (1.04 + 1.81i)25-s + (0.938 + 0.346i)27-s + (0.437 − 0.252i)29-s + (−1.61 − 0.930i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.948 + 0.317i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45180 - 0.236665i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45180 - 0.236665i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (0.683 + 1.59i)T \) |
good | 5 | \( 1 + (-3.40 - 1.96i)T + (2.5 + 4.33i)T^{2} \) |
| 7 | \( 1 + (-0.961 + 0.555i)T + (3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.63 - 2.82i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.124 + 0.216i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + 5.86iT - 17T^{2} \) |
| 19 | \( 1 - 2.19iT - 19T^{2} \) |
| 23 | \( 1 + (-2.79 + 4.83i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.35 + 1.36i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (8.96 + 5.17i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 0.333T + 37T^{2} \) |
| 41 | \( 1 + (5.28 + 3.05i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (8.50 - 4.91i)T + (21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-4.70 - 8.15i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + 4.75iT - 53T^{2} \) |
| 59 | \( 1 + (3.26 - 5.65i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.07 - 1.85i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.501 - 0.289i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 3.26T + 71T^{2} \) |
| 73 | \( 1 + 12.6T + 73T^{2} \) |
| 79 | \( 1 + (-7.67 + 4.42i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (2.34 + 4.05i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 4.75iT - 89T^{2} \) |
| 97 | \( 1 + (-0.916 - 1.58i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.73588690610267680785844637770, −10.85275890536454899893041785293, −9.980145055408674459621752179259, −9.048481867456918169240046426773, −7.52360084777077014957670025186, −6.81943285413148251213596884379, −5.97830299836131637412550025184, −4.90690853033399823089168821747, −2.73487700843428147058709667230, −1.66161821333847984087394560119,
1.59207318195381193775471147935, 3.53616727838665933424054631644, 5.02151739286007470050960688928, 5.60114894132124553514222359991, 6.54658069696898479066805848141, 8.636887172013215134287063467354, 8.948586605480638820261761474821, 9.990475626065776441727375861306, 10.76151928610179656231494511579, 11.75114328855504977143963869636