| L(s) = 1 | + 46.7i·3-s + 1.74e3i·7-s − 2.18e3·9-s − 1.46e4·13-s − 1.65e4i·19-s − 8.14e4·21-s + 7.81e4·25-s − 1.02e5i·27-s + 2.79e5i·31-s − 2.79e5·37-s − 6.83e5i·39-s + 1.24e5i·43-s − 2.21e6·49-s + 7.75e5·57-s + 3.53e6·61-s + ⋯ |
| L(s) = 1 | + 0.999i·3-s + 1.92i·7-s − 9-s − 1.84·13-s − 0.554i·19-s − 1.92·21-s + 25-s − 0.999i·27-s + 1.68i·31-s − 0.907·37-s − 1.84i·39-s + 0.239i·43-s − 2.68·49-s + 0.554·57-s + 1.99·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.3036078164\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3036078164\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 46.7iT \) |
| good | 5 | \( 1 - 7.81e4T^{2} \) |
| 7 | \( 1 - 1.74e3iT - 8.23e5T^{2} \) |
| 11 | \( 1 + 1.94e7T^{2} \) |
| 13 | \( 1 + 1.46e4T + 6.27e7T^{2} \) |
| 17 | \( 1 - 4.10e8T^{2} \) |
| 19 | \( 1 + 1.65e4iT - 8.93e8T^{2} \) |
| 23 | \( 1 + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.72e10T^{2} \) |
| 31 | \( 1 - 2.79e5iT - 2.75e10T^{2} \) |
| 37 | \( 1 + 2.79e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 1.94e11T^{2} \) |
| 43 | \( 1 - 1.24e5iT - 2.71e11T^{2} \) |
| 47 | \( 1 + 5.06e11T^{2} \) |
| 53 | \( 1 - 1.17e12T^{2} \) |
| 59 | \( 1 + 2.48e12T^{2} \) |
| 61 | \( 1 - 3.53e6T + 3.14e12T^{2} \) |
| 67 | \( 1 + 4.90e6iT - 6.06e12T^{2} \) |
| 71 | \( 1 + 9.09e12T^{2} \) |
| 73 | \( 1 - 6.27e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 1.57e5iT - 1.92e13T^{2} \) |
| 83 | \( 1 + 2.71e13T^{2} \) |
| 89 | \( 1 - 4.42e13T^{2} \) |
| 97 | \( 1 + 1.22e7T + 8.07e13T^{2} \) |
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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
−12.06307291823413843129222722742, −10.98533914181917178306603362293, −9.830200720263015215291814085503, −9.117614824531049181911078640840, −8.331233678408138834740670717036, −6.73233544760550273040317385953, −5.31186854528034391930689751108, −4.91067308666795361561061468239, −3.09753818156138453747627976441, −2.28135334293542208660316386341,
0.083591046351202198967947105267, 1.02502713302394980122508437558, 2.41399973116681595484826347764, 3.88789405713229226942239345080, 5.16904429864291205140409461974, 6.73375637921156299919770392406, 7.31895045093175899519348238932, 8.101478860613731474127700911633, 9.659264011391398292781591133079, 10.52472014436010128017524532382
