| L(s) = 1 | − i·3-s − 3.46i·5-s − 3.46·7-s − 9-s − 3.46·15-s + 6·17-s − 4i·19-s + 3.46i·21-s + 6.92·23-s − 6.99·25-s + i·27-s − 3.46i·29-s + 3.46·31-s + 11.9i·35-s + 6.92i·37-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.54i·5-s − 1.30·7-s − 0.333·9-s − 0.894·15-s + 1.45·17-s − 0.917i·19-s + 0.755i·21-s + 1.44·23-s − 1.39·25-s + 0.192i·27-s − 0.643i·29-s + 0.622·31-s + 2.02i·35-s + 1.13i·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.258 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.606131 - 0.789925i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.606131 - 0.789925i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + iT \) |
| good | 5 | \( 1 + 3.46iT - 5T^{2} \) |
| 7 | \( 1 + 3.46T + 7T^{2} \) |
| 11 | \( 1 - 11T^{2} \) |
| 13 | \( 1 - 13T^{2} \) |
| 17 | \( 1 - 6T + 17T^{2} \) |
| 19 | \( 1 + 4iT - 19T^{2} \) |
| 23 | \( 1 - 6.92T + 23T^{2} \) |
| 29 | \( 1 + 3.46iT - 29T^{2} \) |
| 31 | \( 1 - 3.46T + 31T^{2} \) |
| 37 | \( 1 - 6.92iT - 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 - 4iT - 43T^{2} \) |
| 47 | \( 1 - 6.92T + 47T^{2} \) |
| 53 | \( 1 - 3.46iT - 53T^{2} \) |
| 59 | \( 1 + 12iT - 59T^{2} \) |
| 61 | \( 1 - 6.92iT - 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 + 6.92T + 71T^{2} \) |
| 73 | \( 1 - 2T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 2T + 97T^{2} \) |
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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
−12.48267599121218769925128680887, −11.62827665089967695760498063079, −10.02485071710212256049631450457, −9.201938143900970063825023534401, −8.314869136830471831185090861616, −7.08786237725335792325890154552, −5.90217453832004117775645133580, −4.78591559504293675955562833554, −3.10614619104068965326720920865, −0.941758406426872705971917835081,
2.94859128859384174501581603221, 3.62548604982906873751766525295, 5.59151821386866085061786196819, 6.61017659790386139802871915396, 7.52074758091274109258973991389, 9.095635300961643552420937018514, 10.16891608252132565382441305515, 10.50710315024056900055444903250, 11.74748520578607004936442968860, 12.78365828156245111806504779527
