L(s) = 1 | + (0.923 + 0.382i)2-s + (0.707 + 0.707i)4-s + (−1.63 − 1.08i)5-s + (0.382 + 0.923i)8-s + (−0.923 + 0.382i)9-s + (−1.08 − 1.63i)10-s + (1 + i)13-s + i·16-s + (0.923 + 0.382i)17-s − 18-s + (−0.382 − 1.92i)20-s + (1.08 + 2.63i)25-s + (0.541 + 1.30i)26-s + (0.324 + 0.216i)29-s + (−0.382 + 0.923i)32-s + ⋯ |
L(s) = 1 | + (0.923 + 0.382i)2-s + (0.707 + 0.707i)4-s + (−1.63 − 1.08i)5-s + (0.382 + 0.923i)8-s + (−0.923 + 0.382i)9-s + (−1.08 − 1.63i)10-s + (1 + i)13-s + i·16-s + (0.923 + 0.382i)17-s − 18-s + (−0.382 − 1.92i)20-s + (1.08 + 2.63i)25-s + (0.541 + 1.30i)26-s + (0.324 + 0.216i)29-s + (−0.382 + 0.923i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3332 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.336 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.573031907\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.573031907\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.923 - 0.382i)T \) |
| 7 | \( 1 \) |
| 17 | \( 1 + (-0.923 - 0.382i)T \) |
good | 3 | \( 1 + (0.923 - 0.382i)T^{2} \) |
| 5 | \( 1 + (1.63 + 1.08i)T + (0.382 + 0.923i)T^{2} \) |
| 11 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 13 | \( 1 + (-1 - i)T + iT^{2} \) |
| 19 | \( 1 + (0.707 + 0.707i)T^{2} \) |
| 23 | \( 1 + (-0.923 - 0.382i)T^{2} \) |
| 29 | \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \) |
| 31 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 37 | \( 1 + (-1.92 + 0.382i)T + (0.923 - 0.382i)T^{2} \) |
| 41 | \( 1 + (0.923 - 0.617i)T + (0.382 - 0.923i)T^{2} \) |
| 43 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 47 | \( 1 - iT^{2} \) |
| 53 | \( 1 + (0.707 + 0.292i)T + (0.707 + 0.707i)T^{2} \) |
| 59 | \( 1 + (-0.707 + 0.707i)T^{2} \) |
| 61 | \( 1 + (-0.923 - 1.38i)T + (-0.382 + 0.923i)T^{2} \) |
| 67 | \( 1 + T^{2} \) |
| 71 | \( 1 + (-0.923 + 0.382i)T^{2} \) |
| 73 | \( 1 + (-0.324 - 0.216i)T + (0.382 + 0.923i)T^{2} \) |
| 79 | \( 1 + (0.923 + 0.382i)T^{2} \) |
| 83 | \( 1 + (-0.707 - 0.707i)T^{2} \) |
| 89 | \( 1 + (1.41 - 1.41i)T - iT^{2} \) |
| 97 | \( 1 + (-0.216 + 0.324i)T + (-0.382 - 0.923i)T^{2} \) |
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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
−8.480634707227969672555322845747, −8.229407852401658030019127049983, −7.56815178776719940674534589642, −6.66844379626575405626466724302, −5.76827835977077860137970570237, −5.06719832626920243790083460975, −4.25811343173399855974042942822, −3.76439734420729673774313673837, −2.87679124674465301627324536735, −1.35321873765316198875998271096,
0.76665215685558266321817816891, 2.64396345573457361494595660662, 3.27088980362568678000930031419, 3.67912524995280702781805175386, 4.63986592546920063586461606090, 5.67711682678614953076162788228, 6.32087431685398055533573979630, 7.05733558542951134285153344298, 7.930011972751165191526102376796, 8.298793600213641171997144208222