| L(s) = 1 | − 1.08·3-s − 2.09i·5-s − 3.72·7-s − 7.81·9-s − 6.64i·11-s − 9.32·13-s + 2.27i·15-s − 2.02·17-s + (−17.3 + 7.78i)19-s + 4.04·21-s − 7.73·23-s + 20.6·25-s + 18.2·27-s + 35.3·29-s + 37.0i·31-s + ⋯ |
| L(s) = 1 | − 0.362·3-s − 0.418i·5-s − 0.531·7-s − 0.868·9-s − 0.603i·11-s − 0.717·13-s + 0.151i·15-s − 0.118·17-s + (−0.912 + 0.409i)19-s + 0.192·21-s − 0.336·23-s + 0.824·25-s + 0.677·27-s + 1.21·29-s + 1.19i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.775 - 0.631i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9179610307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9179610307\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (17.3 - 7.78i)T \) |
| good | 3 | \( 1 + 1.08T + 9T^{2} \) |
| 5 | \( 1 + 2.09iT - 25T^{2} \) |
| 7 | \( 1 + 3.72T + 49T^{2} \) |
| 11 | \( 1 + 6.64iT - 121T^{2} \) |
| 13 | \( 1 + 9.32T + 169T^{2} \) |
| 17 | \( 1 + 2.02T + 289T^{2} \) |
| 23 | \( 1 + 7.73T + 529T^{2} \) |
| 29 | \( 1 - 35.3T + 841T^{2} \) |
| 31 | \( 1 - 37.0iT - 961T^{2} \) |
| 37 | \( 1 - 33.2T + 1.36e3T^{2} \) |
| 41 | \( 1 + 57.3iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 28.3iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 73.9T + 2.20e3T^{2} \) |
| 53 | \( 1 - 59.7T + 2.80e3T^{2} \) |
| 59 | \( 1 - 60.0T + 3.48e3T^{2} \) |
| 61 | \( 1 - 18.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 91.3T + 4.48e3T^{2} \) |
| 71 | \( 1 - 83.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 10.2T + 5.32e3T^{2} \) |
| 79 | \( 1 - 58.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 - 5.17iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 145. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 39.3iT - 9.40e3T^{2} \) |
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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
−9.678241609985031837753214880770, −8.595123821145739706398874230990, −8.333765193699660734063086277316, −6.96184477798373777378531649624, −6.28370917950469442636874654757, −5.41969747986879792770591731368, −4.61023488557858989507081787677, −3.39131123365269215540093470904, −2.42253563350664529774126111701, −0.77189355059486204177643120064,
0.40329086922719113486477303146, 2.27272108278104165745157524505, 3.05498323246608796374948441756, 4.36478707255600461949375558103, 5.18146628873350150829482447830, 6.34997612510582085043801761791, 6.69091629478855345279642061232, 7.82826489000191891542816278692, 8.617737785413711212396691935577, 9.622445339940424610218442094577
