| L(s) = 1 | − 1.30·3-s + 2.23i·5-s + 4.63i·7-s − 7.29·9-s + 16.1i·11-s + 2.26·13-s − 2.92i·15-s + 24.7i·17-s + 21.2i·19-s − 6.05i·21-s + (−10.0 − 20.6i)23-s − 5.00·25-s + 21.2·27-s − 40.7·29-s + 52.4·31-s + ⋯ |
| L(s) = 1 | − 0.435·3-s + 0.447i·5-s + 0.661i·7-s − 0.810·9-s + 1.46i·11-s + 0.174·13-s − 0.194i·15-s + 1.45i·17-s + 1.11i·19-s − 0.288i·21-s + (−0.437 − 0.899i)23-s − 0.200·25-s + 0.788·27-s − 1.40·29-s + 1.69·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6849905285\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6849905285\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23iT \) |
| 23 | \( 1 + (10.0 + 20.6i)T \) |
| good | 3 | \( 1 + 1.30T + 9T^{2} \) |
| 7 | \( 1 - 4.63iT - 49T^{2} \) |
| 11 | \( 1 - 16.1iT - 121T^{2} \) |
| 13 | \( 1 - 2.26T + 169T^{2} \) |
| 17 | \( 1 - 24.7iT - 289T^{2} \) |
| 19 | \( 1 - 21.2iT - 361T^{2} \) |
| 29 | \( 1 + 40.7T + 841T^{2} \) |
| 31 | \( 1 - 52.4T + 961T^{2} \) |
| 37 | \( 1 - 18.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 50.9T + 1.68e3T^{2} \) |
| 43 | \( 1 - 23.7iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 58.3T + 2.20e3T^{2} \) |
| 53 | \( 1 + 54.3iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 21.2T + 3.48e3T^{2} \) |
| 61 | \( 1 + 68.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 57.6iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 34.8T + 5.04e3T^{2} \) |
| 73 | \( 1 + 73.9T + 5.32e3T^{2} \) |
| 79 | \( 1 - 43.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 63.2iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 37.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 38.9iT - 9.40e3T^{2} \) |
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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
−9.694290987931540495461013751159, −8.534548837241460295004698734941, −8.120238339659437048608974443919, −7.02160067071085910702226358102, −6.19554688073033454517804198942, −5.69975529037443941894060365959, −4.64843022349130821279182750829, −3.72670500829291162852000012575, −2.54625671896354242462105476121, −1.67799645110423590625974428660,
0.22307756628319840804499674333, 0.961357304760583183961440483707, 2.64950518371254948534481214885, 3.51086794610821875574416607565, 4.60210302199530102941588404707, 5.47219408902464808849748420198, 6.02585499505058486430248244056, 7.06991042820086856641301063614, 7.79571062484477234060327406405, 8.817387954817624561114247673116
