| L(s) = 1 | − 2.89i·2-s − 3.45i·3-s + 23.6·4-s − 9.99·6-s + 148. i·7-s − 160. i·8-s + 231.·9-s − 712.·11-s − 81.7i·12-s + 169i·13-s + 428.·14-s + 290.·16-s + 1.13e3i·17-s − 668. i·18-s − 1.40e3·19-s + ⋯ |
| L(s) = 1 | − 0.511i·2-s − 0.221i·3-s + 0.738·4-s − 0.113·6-s + 1.14i·7-s − 0.888i·8-s + 0.950·9-s − 1.77·11-s − 0.163i·12-s + 0.277i·13-s + 0.584·14-s + 0.284·16-s + 0.949i·17-s − 0.486i·18-s − 0.889·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 325 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7359125028\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7359125028\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 13 | \( 1 - 169iT \) |
| good | 2 | \( 1 + 2.89iT - 32T^{2} \) |
| 3 | \( 1 + 3.45iT - 243T^{2} \) |
| 7 | \( 1 - 148. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 712.T + 1.61e5T^{2} \) |
| 17 | \( 1 - 1.13e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.40e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 897. iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 3.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 7.97e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.97e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.55e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.30e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 7.60e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.41e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 4.98e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.51e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.85e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 6.80e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.13e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.98e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.38e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 8.92e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.47e5iT - 8.58e9T^{2} \) |
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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
−10.94307231824655215935111506083, −10.42090738740085153348332543837, −9.368542857421231667108198194735, −8.154552571186677808607664819490, −7.29704627487840507181404034676, −6.20029091804524294991739569841, −5.22660032675330831977259968577, −3.69371101474690230104215724911, −2.37580094412295084864019564854, −1.74202425904293740801679326167,
0.16306908020263547960215966687, 1.83509203596215331450287927992, 3.15603081753176675287243664753, 4.57051091219701523062830980597, 5.51178108220517065197003432970, 6.88779430140820356208103648068, 7.42301633187027906749652171299, 8.222496831202450606822491048662, 9.761041995205028206277629860463, 10.64277100467776110643572678749
