| L(s) = 1 | + 2-s + 4-s − 3.02i·5-s − 7-s + 8-s − 3.02i·10-s + 4.70i·11-s + 6.26i·13-s − 14-s + 16-s + 4.70i·17-s + (4.28 + 0.781i)19-s − 3.02i·20-s + 4.70i·22-s − 0.460i·23-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.35i·5-s − 0.377·7-s + 0.353·8-s − 0.958i·10-s + 1.41i·11-s + 1.73i·13-s − 0.267·14-s + 0.250·16-s + 1.14i·17-s + (0.983 + 0.179i)19-s − 0.677i·20-s + 1.00i·22-s − 0.0961i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2394 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.489798184\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.489798184\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 19 | \( 1 + (-4.28 - 0.781i)T \) |
| good | 5 | \( 1 + 3.02iT - 5T^{2} \) |
| 11 | \( 1 - 4.70iT - 11T^{2} \) |
| 13 | \( 1 - 6.26iT - 13T^{2} \) |
| 17 | \( 1 - 4.70iT - 17T^{2} \) |
| 23 | \( 1 + 0.460iT - 23T^{2} \) |
| 29 | \( 1 + 3.18T + 29T^{2} \) |
| 31 | \( 1 - 7.73iT - 31T^{2} \) |
| 37 | \( 1 + 1.81iT - 37T^{2} \) |
| 41 | \( 1 + 8.40T + 41T^{2} \) |
| 43 | \( 1 - 4.86T + 43T^{2} \) |
| 47 | \( 1 + 12.5iT - 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 5.46T + 61T^{2} \) |
| 67 | \( 1 - 12.0iT - 67T^{2} \) |
| 71 | \( 1 - 2.20T + 71T^{2} \) |
| 73 | \( 1 + 13.2T + 73T^{2} \) |
| 79 | \( 1 - 8.69iT - 79T^{2} \) |
| 83 | \( 1 + 8.37iT - 83T^{2} \) |
| 89 | \( 1 - 6.19T + 89T^{2} \) |
| 97 | \( 1 - 9.50iT - 97T^{2} \) |
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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
−8.979150519685630082718890934807, −8.461989075000494696539375775992, −7.21194830913141580559756852407, −6.85821795983096420192905451373, −5.71001540487746021056745880538, −5.01686255076832949843020308953, −4.29401715063917770286757630769, −3.68525196716369426334309966470, −2.11356616347057732663667236254, −1.41759859151982357637974255977,
0.67144243366547408297426278110, 2.62056850513009256418665579476, 3.09291762552602441867259191693, 3.66388139030869027203165679676, 5.09263707378931210368676367521, 5.82293115377265795779281439468, 6.33039988282156195435989905321, 7.41706961719824443645916930646, 7.68811900177132047638824799613, 8.855423743732890294863720687555
