| L(s) = 1 | + 73.9i·5-s + 112.·7-s + 575. i·11-s − 117. i·13-s + 223.·17-s + 1.75e3i·19-s + 2.36e3·23-s − 2.34e3·25-s − 3.86e3i·29-s + 1.59e3·31-s + 8.33e3i·35-s − 4.73e3i·37-s − 8.15e3·41-s + 4.92e3i·43-s − 2.10e4·47-s + ⋯ |
| L(s) = 1 | + 1.32i·5-s + 0.869·7-s + 1.43i·11-s − 0.193i·13-s + 0.187·17-s + 1.11i·19-s + 0.930·23-s − 0.750·25-s − 0.853i·29-s + 0.297·31-s + 1.15i·35-s − 0.568i·37-s − 0.757·41-s + 0.405i·43-s − 1.39·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.666 - 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.906825876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.906825876\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 - 73.9iT - 3.12e3T^{2} \) |
| 7 | \( 1 - 112.T + 1.68e4T^{2} \) |
| 11 | \( 1 - 575. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 117. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 223.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.75e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 2.36e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 3.86e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 1.59e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 4.73e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 8.15e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.92e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.10e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.27e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.41e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 - 4.20e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 5.41e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 4.38e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.02e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 4.37e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 6.44e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 6.23e4T + 8.58e9T^{2} \) |
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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
−11.26067956501859052248184691166, −10.36104433762695947101427960218, −9.707677973087422992148568227120, −8.221369700145145211813844998834, −7.38972688672310205566512998695, −6.57683677947969463346714252115, −5.26649295916102066225488697585, −4.07638563962504100767418256542, −2.73801624779992579458299459931, −1.59980091469193879661132269133,
0.53165674816913544331693987421, 1.50303773501428859636856697214, 3.23027113170659459449867045506, 4.74178007411790854998854416100, 5.26305641055301408736743077913, 6.62526604134952859068826132782, 8.055421777994481797335486752037, 8.639575829049295757144312262545, 9.400835539203121263593938980059, 10.88171391089330536346112034658
