| L(s) = 1 | + 3.46i·3-s + 20.7·5-s − 40i·7-s + 69·9-s − 197. i·11-s − 256.·13-s + 72i·15-s − 198·17-s + 252. i·19-s + 138.·21-s − 504i·23-s − 193·25-s + 519. i·27-s + 436.·29-s − 1.69e3i·31-s + ⋯ |
| L(s) = 1 | + 0.384i·3-s + 0.831·5-s − 0.816i·7-s + 0.851·9-s − 1.63i·11-s − 1.51·13-s + 0.320i·15-s − 0.685·17-s + 0.700i·19-s + 0.314·21-s − 0.952i·23-s − 0.308·25-s + 0.712i·27-s + 0.518·29-s − 1.76i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.704334383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.704334383\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 - 3.46iT - 81T^{2} \) |
| 5 | \( 1 - 20.7T + 625T^{2} \) |
| 7 | \( 1 + 40iT - 2.40e3T^{2} \) |
| 11 | \( 1 + 197. iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 256.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 198T + 8.35e4T^{2} \) |
| 19 | \( 1 - 252. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 504iT - 2.79e5T^{2} \) |
| 29 | \( 1 - 436.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.69e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.75e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 18T + 2.82e6T^{2} \) |
| 43 | \( 1 - 1.54e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.44e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 3.26e3T + 7.89e6T^{2} \) |
| 59 | \( 1 + 5.26e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 1.00e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 1.80e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 7.84e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.31e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.14e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 9.09e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 + 3.11e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 4.73e3T + 8.85e7T^{2} \) |
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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
−10.84803181534786452141513043708, −10.16239271052879557810761414081, −9.451176943294348562833142081757, −8.222092825381766641031206322116, −7.07973087495085395518413993831, −6.04079928578770910766088824578, −4.84420081268402590415174651016, −3.72626744247880776952472586155, −2.17887083965916783074699847701, −0.52384444794500879738276417175,
1.71348643506054050584505671583, 2.47976770141170609880179784715, 4.54151124573490609705817362297, 5.39257586515213976395853806830, 6.86886031215732061519030224308, 7.32482539274591871620990500582, 8.911015452755834288785577988780, 9.730561435120599827754438549429, 10.35404186941721977214158933708, 12.01477354566105014958036209575
