| L(s) = 1 | + 16.0·3-s − 100.·5-s + 99.5·7-s + 15.6·9-s − 121·11-s + 1.12e3·13-s − 1.61e3·15-s + 5.49·17-s − 576.·19-s + 1.60e3·21-s − 961.·23-s + 7.00e3·25-s − 3.65e3·27-s − 2.01e3·29-s + 5.68e3·31-s − 1.94e3·33-s − 1.00e4·35-s − 7.36e3·37-s + 1.81e4·39-s − 1.89e4·41-s − 2.19e4·43-s − 1.57e3·45-s − 2.00e4·47-s − 6.89e3·49-s + 88.4·51-s − 9.83e3·53-s + 1.21e4·55-s + ⋯ |
| L(s) = 1 | + 1.03·3-s − 1.80·5-s + 0.767·7-s + 0.0645·9-s − 0.301·11-s + 1.85·13-s − 1.85·15-s + 0.00461·17-s − 0.366·19-s + 0.792·21-s − 0.379·23-s + 2.24·25-s − 0.965·27-s − 0.443·29-s + 1.06·31-s − 0.311·33-s − 1.38·35-s − 0.884·37-s + 1.91·39-s − 1.75·41-s − 1.81·43-s − 0.116·45-s − 1.32·47-s − 0.410·49-s + 0.00476·51-s − 0.480·53-s + 0.542·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 352 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + 121T \) |
| good | 3 | \( 1 - 16.0T + 243T^{2} \) |
| 5 | \( 1 + 100.T + 3.12e3T^{2} \) |
| 7 | \( 1 - 99.5T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.12e3T + 3.71e5T^{2} \) |
| 17 | \( 1 - 5.49T + 1.41e6T^{2} \) |
| 19 | \( 1 + 576.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 961.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.01e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.68e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.36e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.89e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.19e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 2.00e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 9.83e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.36e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.86e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.26e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.41e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.60e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.15e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.29e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 1.04e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.28e4T + 8.58e9T^{2} \) |
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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
−10.29379446898635908744221911163, −8.635891986447211059577685489274, −8.404613135841374290886900104451, −7.77706748048000589929834167792, −6.55703506169688009237356789941, −4.93108720143484759980563141157, −3.79455246898175508514489406624, −3.23405437653200203906764930752, −1.57435494228360347865445976199, 0,
1.57435494228360347865445976199, 3.23405437653200203906764930752, 3.79455246898175508514489406624, 4.93108720143484759980563141157, 6.55703506169688009237356789941, 7.77706748048000589929834167792, 8.404613135841374290886900104451, 8.635891986447211059577685489274, 10.29379446898635908744221911163
