| L(s) = 1 | − 5.13·3-s + 6.19i·7-s + 17.3·9-s + 20.0·11-s − 15.8i·13-s + 6.98·17-s + 10.3·19-s − 31.7i·21-s + 22.3i·23-s − 42.9·27-s − 4.20i·29-s − 20.7i·31-s − 102.·33-s − 35.4i·37-s + 81.4i·39-s + ⋯ |
| L(s) = 1 | − 1.71·3-s + 0.884i·7-s + 1.92·9-s + 1.82·11-s − 1.22i·13-s + 0.411·17-s + 0.546·19-s − 1.51i·21-s + 0.973i·23-s − 1.58·27-s − 0.145i·29-s − 0.668i·31-s − 3.11·33-s − 0.958i·37-s + 2.08i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1600 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.187493974\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187493974\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 5.13T + 9T^{2} \) |
| 7 | \( 1 - 6.19iT - 49T^{2} \) |
| 11 | \( 1 - 20.0T + 121T^{2} \) |
| 13 | \( 1 + 15.8iT - 169T^{2} \) |
| 17 | \( 1 - 6.98T + 289T^{2} \) |
| 19 | \( 1 - 10.3T + 361T^{2} \) |
| 23 | \( 1 - 22.3iT - 529T^{2} \) |
| 29 | \( 1 + 4.20iT - 841T^{2} \) |
| 31 | \( 1 + 20.7iT - 961T^{2} \) |
| 37 | \( 1 + 35.4iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 37.0T + 1.68e3T^{2} \) |
| 43 | \( 1 + 23.8T + 1.84e3T^{2} \) |
| 47 | \( 1 - 48.7iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 77.4iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 0.497T + 3.48e3T^{2} \) |
| 61 | \( 1 - 60.7iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 82.5T + 4.48e3T^{2} \) |
| 71 | \( 1 - 28.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 10.1T + 5.32e3T^{2} \) |
| 79 | \( 1 + 87.5iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 103.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 49.2T + 7.92e3T^{2} \) |
| 97 | \( 1 + 84.4T + 9.40e3T^{2} \) |
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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
−9.459373602153349581548949071234, −8.411000027484179364418305580134, −7.33158119486880240856869839038, −6.58052535153739982725130323286, −5.68156611827710038074371041081, −5.49878536539668992121097684291, −4.32760750322741265708753684036, −3.31714760690643435217129528072, −1.64301756729230576465752833880, −0.62922693846647221252834353971,
0.828385154765123476446720462711, 1.57671782548314278027919547990, 3.64578149426544338316491186328, 4.37194599695209720978210863123, 5.08346347688628276026562766534, 6.21417230061952592521409440732, 6.74270262777211355925274677837, 7.15001286229127651014125103910, 8.533443828973204969749333211124, 9.496978513187910536233274856667
