| L(s) = 1 | − 2.82·2-s + 8.00·4-s − 0.710i·5-s − 22.6·8-s + 2.01i·10-s − 151.·11-s + 260. i·13-s + 64.0·16-s + 385. i·17-s + 390. i·19-s − 5.68i·20-s + 428.·22-s + 177.·23-s + 624.·25-s − 737. i·26-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.500·4-s − 0.0284i·5-s − 0.353·8-s + 0.0201i·10-s − 1.25·11-s + 1.54i·13-s + 0.250·16-s + 1.33i·17-s + 1.08i·19-s − 0.0142i·20-s + 0.886·22-s + 0.335·23-s + 0.999·25-s − 1.09i·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.912 + 0.409i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.2311998832\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2311998832\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2.82T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.710iT - 625T^{2} \) |
| 11 | \( 1 + 151.T + 1.46e4T^{2} \) |
| 13 | \( 1 - 260. iT - 2.85e4T^{2} \) |
| 17 | \( 1 - 385. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 390. iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 177.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 320.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.34e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 797.T + 1.87e6T^{2} \) |
| 41 | \( 1 + 815. iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 2.16e3T + 3.41e6T^{2} \) |
| 47 | \( 1 - 4.28e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.17e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 4.70e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 2.53e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 + 4.09e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 2.25e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + 4.65e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 4.19e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 7.79e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 9.46e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 - 9.94e3iT - 8.85e7T^{2} \) |
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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
−10.06761523971637973526847264086, −9.172836309541157981681816346735, −8.355298529187544875976439236523, −7.70188752908071278503395344238, −6.69238886922514729614678206804, −5.91462660514995331153210304986, −4.76041735526901602208643096786, −3.64917762434786145287767941705, −2.36011147685952423187654662312, −1.44522177720359180552738795983,
0.07571366206779459053786680440, 0.958869225167929989307498791344, 2.62697420434530297063870147519, 3.10647737106999495170724098661, 4.96324594642556271356790913802, 5.41453347474129537843072411074, 6.86201589544951280512520310694, 7.37561166091732352838327810715, 8.394598632800526980607825397573, 8.921667139731876803183748338969
