| L(s) = 1 | − 4·2-s + 16·4-s − 75.4·5-s − 64·8-s + 301.·10-s + 149.·11-s − 349.·13-s + 256·16-s − 1.14e3·17-s + 2.79e3·19-s − 1.20e3·20-s − 597.·22-s − 1.81e3·23-s + 2.57e3·25-s + 1.39e3·26-s + 759.·29-s − 9.03e3·31-s − 1.02e3·32-s + 4.59e3·34-s + 7.79e3·37-s − 1.11e4·38-s + 4.83e3·40-s + 7.64e3·41-s + 1.21e4·43-s + 2.39e3·44-s + 7.25e3·46-s + 2.45e4·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 1.35·5-s − 0.353·8-s + 0.954·10-s + 0.372·11-s − 0.573·13-s + 0.250·16-s − 0.964·17-s + 1.77·19-s − 0.675·20-s − 0.263·22-s − 0.715·23-s + 0.823·25-s + 0.405·26-s + 0.167·29-s − 1.68·31-s − 0.176·32-s + 0.682·34-s + 0.936·37-s − 1.25·38-s + 0.477·40-s + 0.709·41-s + 1.00·43-s + 0.186·44-s + 0.505·46-s + 1.62·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 75.4T + 3.12e3T^{2} \) |
| 11 | \( 1 - 149.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 349.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.14e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.79e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.81e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 759.T + 2.05e7T^{2} \) |
| 31 | \( 1 + 9.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 7.79e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 7.64e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.21e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.45e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.35e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.63e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.53e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 5.43e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 7.01e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 4.44e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 6.16e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.71e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.85e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.23e4T + 8.58e9T^{2} \) |
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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
−9.122967579408345301105672359664, −7.87488213768432862565109879127, −7.59986802901357946926291486359, −6.70346829860673560579288455765, −5.51679574437906724904096269878, −4.33068137825321978997266489929, −3.50311526236590679312837458846, −2.33876026178801805805615635408, −0.946722620384081625664183226747, 0,
0.946722620384081625664183226747, 2.33876026178801805805615635408, 3.50311526236590679312837458846, 4.33068137825321978997266489929, 5.51679574437906724904096269878, 6.70346829860673560579288455765, 7.59986802901357946926291486359, 7.87488213768432862565109879127, 9.122967579408345301105672359664
