| L(s) = 1 | − 2.99i·3-s − 0.369i·5-s − 7-s − 5.96·9-s + 3.93i·11-s + 6.07i·13-s − 1.10·15-s + 6.75·17-s − 0.117i·19-s + 2.99i·21-s − 8.67·23-s + 4.86·25-s + 8.86i·27-s − 6.27i·29-s − 0.808·31-s + ⋯ |
| L(s) = 1 | − 1.72i·3-s − 0.165i·5-s − 0.377·7-s − 1.98·9-s + 1.18i·11-s + 1.68i·13-s − 0.285·15-s + 1.63·17-s − 0.0268i·19-s + 0.653i·21-s − 1.80·23-s + 0.972·25-s + 1.70i·27-s − 1.16i·29-s − 0.145·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.627528323\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.627528323\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 + 2.99iT - 3T^{2} \) |
| 5 | \( 1 + 0.369iT - 5T^{2} \) |
| 11 | \( 1 - 3.93iT - 11T^{2} \) |
| 13 | \( 1 - 6.07iT - 13T^{2} \) |
| 17 | \( 1 - 6.75T + 17T^{2} \) |
| 19 | \( 1 + 0.117iT - 19T^{2} \) |
| 23 | \( 1 + 8.67T + 23T^{2} \) |
| 29 | \( 1 + 6.27iT - 29T^{2} \) |
| 31 | \( 1 + 0.808T + 31T^{2} \) |
| 37 | \( 1 + 10.1iT - 37T^{2} \) |
| 41 | \( 1 - 7.27T + 41T^{2} \) |
| 43 | \( 1 + 0.0365iT - 43T^{2} \) |
| 47 | \( 1 - 4.06T + 47T^{2} \) |
| 53 | \( 1 + 2.53iT - 53T^{2} \) |
| 59 | \( 1 + 5.53iT - 59T^{2} \) |
| 61 | \( 1 + 6.02iT - 61T^{2} \) |
| 67 | \( 1 + 4.88iT - 67T^{2} \) |
| 71 | \( 1 + 2.35T + 71T^{2} \) |
| 73 | \( 1 - 6.82T + 73T^{2} \) |
| 79 | \( 1 - 6.35T + 79T^{2} \) |
| 83 | \( 1 + 8.65iT - 83T^{2} \) |
| 89 | \( 1 - 3.56T + 89T^{2} \) |
| 97 | \( 1 + 0.211T + 97T^{2} \) |
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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
−8.004581551905017774709536407213, −7.61628499748497821261826383117, −6.95509425162501452845984224273, −6.29027367005761381812172463974, −5.69091971688638051903291665698, −4.53663364163108529490250934146, −3.61764893501990931520262966605, −2.23709121431175003191455375077, −1.90235733347402017825022569809, −0.67992942416339925324274308680,
0.820490165856555578728546576658, 2.95933136875817948009928066171, 3.19961212365682945581416512127, 3.97670971594425190417264618931, 5.00686649697336578940354936731, 5.71173141220682472131217231636, 6.02638481011636600546667573986, 7.45142458697631624544862451790, 8.290446796369419169504283098650, 8.695954071760758188493122992886
