| L(s) = 1 | − 40.2i·5-s + 14.7·7-s + 81.6i·11-s + 278.·13-s + 10.8i·17-s + 532.·19-s − 810. i·23-s − 996.·25-s + 297. i·29-s + 195.·31-s − 594. i·35-s − 2.09e3·37-s − 1.56e3i·41-s − 92.1·43-s − 2.13e3i·47-s + ⋯ |
| L(s) = 1 | − 1.61i·5-s + 0.301·7-s + 0.674i·11-s + 1.64·13-s + 0.0376i·17-s + 1.47·19-s − 1.53i·23-s − 1.59·25-s + 0.353i·29-s + 0.203·31-s − 0.485i·35-s − 1.53·37-s − 0.933i·41-s − 0.0498·43-s − 0.966i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 324 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(2.125041401\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.125041401\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 40.2iT - 625T^{2} \) |
| 7 | \( 1 - 14.7T + 2.40e3T^{2} \) |
| 11 | \( 1 - 81.6iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 278.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 10.8iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 532.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 810. iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 297. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 195.T + 9.23e5T^{2} \) |
| 37 | \( 1 + 2.09e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.56e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 92.1T + 3.41e6T^{2} \) |
| 47 | \( 1 + 2.13e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 2.57e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.55e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 5.37e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 915.T + 2.01e7T^{2} \) |
| 71 | \( 1 + 8.21e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.43e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 4.63e3T + 3.89e7T^{2} \) |
| 83 | \( 1 - 5.99e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 8.43e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 6.03e3T + 8.85e7T^{2} \) |
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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
−10.76320567592237876030645563393, −9.665966120302574378847112377010, −8.699825727885142434442880421553, −8.240394357119978203336858203737, −6.88035659167040648293494017944, −5.54935560485916566347228213373, −4.78361839457394050626156858059, −3.66909700850142729990495161114, −1.72933031917667954452690632132, −0.71846396908406444568713997372,
1.36096554379665878650283967724, 3.03538464029662912114747700894, 3.67958903342371623337704307565, 5.50882606717578649304853724614, 6.36197188468458551666609435685, 7.33868762590800659111533538354, 8.240279999048629040534021499418, 9.465132413659769292429925626345, 10.45063588007574472009860570274, 11.30445231531199316182676425209
