| L(s) = 1 | + 11.1i·3-s + (5 − 55.6i)5-s − 122. i·7-s + 119.·9-s − 100·11-s + 734. i·13-s + (619. + 55.6i)15-s − 979. i·17-s + 2.24e3·19-s + 1.36e3·21-s − 3.41e3i·23-s + (−3.07e3 − 556. i)25-s + 4.03e3i·27-s − 7.85e3·29-s + 2.14e3·31-s + ⋯ |
| L(s) = 1 | + 0.714i·3-s + (0.0894 − 0.995i)5-s − 0.944i·7-s + 0.489·9-s − 0.249·11-s + 1.20i·13-s + (0.711 + 0.0638i)15-s − 0.822i·17-s + 1.42·19-s + 0.674·21-s − 1.34i·23-s + (−0.983 − 0.178i)25-s + 1.06i·27-s − 1.73·29-s + 0.400·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.0894 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.650248077\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.650248077\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-5 + 55.6i)T \) |
| good | 3 | \( 1 - 11.1iT - 243T^{2} \) |
| 7 | \( 1 + 122. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 100T + 1.61e5T^{2} \) |
| 13 | \( 1 - 734. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 979. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.24e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.41e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 7.85e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.14e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.04e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 7.41e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.77e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 9.43e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.42e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.59e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.05e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.87e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 3.76e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.40e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 7.97e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.62e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 826T + 5.58e9T^{2} \) |
| 97 | \( 1 - 3.75e4iT - 8.58e9T^{2} \) |
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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
−10.36073297503395336830007795927, −9.621212577586265929609035630620, −8.936388524099825809113371961115, −7.66247122572529922476196792655, −6.78003575548190241237761128212, −5.19399462906579573142919958004, −4.52887319864426694364326652379, −3.59094956574471590014844025913, −1.70459240082522058207036972744, −0.44340848046082226308189162315,
1.37045245245271877766990722539, 2.56114965143684325321073482418, 3.59536784670161724108992805416, 5.46181142286028004731511694339, 6.10064657929294140039396587696, 7.45625845686601697938979634600, 7.77157959699295499897780676075, 9.271758284613131533267663081524, 10.12581781821141615021545329504, 11.11574585690471616448469564112
