L(s) = 1 | + (−7.90 + 13.6i)5-s + (15.2 − 10.5i)7-s + (−15.1 − 26.1i)11-s − 61.6·13-s + (28.0 + 48.6i)17-s + (69.7 − 120. i)19-s + (−4.07 + 7.05i)23-s + (−62.5 − 108. i)25-s + 0.217·29-s + (−88.2 − 152. i)31-s + (24.5 + 291. i)35-s + (−105. + 182. i)37-s + 293.·41-s + 434.·43-s + (241. − 418. i)47-s + ⋯ |
L(s) = 1 | + (−0.707 + 1.22i)5-s + (0.820 − 0.570i)7-s + (−0.414 − 0.718i)11-s − 1.31·13-s + (0.400 + 0.693i)17-s + (0.841 − 1.45i)19-s + (−0.0369 + 0.0639i)23-s + (−0.500 − 0.866i)25-s + 0.00139·29-s + (−0.511 − 0.885i)31-s + (0.118 + 1.40i)35-s + (−0.468 + 0.812i)37-s + 1.11·41-s + 1.54·43-s + (0.750 − 1.29i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.621 + 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.368012556\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.368012556\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-15.2 + 10.5i)T \) |
good | 5 | \( 1 + (7.90 - 13.6i)T + (-62.5 - 108. i)T^{2} \) |
| 11 | \( 1 + (15.1 + 26.1i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + 61.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + (-28.0 - 48.6i)T + (-2.45e3 + 4.25e3i)T^{2} \) |
| 19 | \( 1 + (-69.7 + 120. i)T + (-3.42e3 - 5.94e3i)T^{2} \) |
| 23 | \( 1 + (4.07 - 7.05i)T + (-6.08e3 - 1.05e4i)T^{2} \) |
| 29 | \( 1 - 0.217T + 2.43e4T^{2} \) |
| 31 | \( 1 + (88.2 + 152. i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + (105. - 182. i)T + (-2.53e4 - 4.38e4i)T^{2} \) |
| 41 | \( 1 - 293.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 434.T + 7.95e4T^{2} \) |
| 47 | \( 1 + (-241. + 418. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + (-10.2 - 17.7i)T + (-7.44e4 + 1.28e5i)T^{2} \) |
| 59 | \( 1 + (115. + 200. i)T + (-1.02e5 + 1.77e5i)T^{2} \) |
| 61 | \( 1 + (-419. + 726. i)T + (-1.13e5 - 1.96e5i)T^{2} \) |
| 67 | \( 1 + (-312. - 540. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + 227.T + 3.57e5T^{2} \) |
| 73 | \( 1 + (21.5 + 37.2i)T + (-1.94e5 + 3.36e5i)T^{2} \) |
| 79 | \( 1 + (-154. + 267. i)T + (-2.46e5 - 4.26e5i)T^{2} \) |
| 83 | \( 1 + 1.23e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + (-572. + 991. i)T + (-3.52e5 - 6.10e5i)T^{2} \) |
| 97 | \( 1 - 1.68e3T + 9.12e5T^{2} \) |
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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
−10.60421870280423488717947458062, −9.662927791418761587038870976405, −8.368767621732277562971573804592, −7.43679522211638921073382501573, −7.11289752210983076815205894352, −5.67714436043941279817091190075, −4.56459608577291561350087214122, −3.43068861999978657781875115732, −2.40728015745812443903083967261, −0.49453654455018106161126436663,
1.10020835092450792240090739989, 2.45851248802879476470461245531, 4.11047694961148195491266772120, 5.00237232027300811430043188168, 5.57928343102369630854293781176, 7.55269286661602630842886772568, 7.68781872967650208935017756561, 8.890210550233631075996437783813, 9.561908048664167091693306064876, 10.64299757660711245141546290331